First, the overall subject verses are below (for easy reference); then I will compare relevant Old Testament verses which point to the future New Testament fulfillment.

The two Goats: Lev. 16

And he shall take the two goats, and present them before the LORD at the door of the tabernacle of the congregation. And Aaron shall cast lots upon the two goats; one lot for the LORD, and the other lot for the scapegoat. And Aaron shall bring the goat upon which the LORD'S lot fell, and offer him for a sin offering. But the goat, on which the lot fell to be the scapegoat, shall be presented alive before the LORD, to make an atonement with him, and to let him go for a scapegoat into the wilderness. And Aaron shall bring the bullock of the sin offering, which is for himself, and shall make an atonement for himself, and for his house, and shall kill the bullock of the sin offering which is for himself: And he shall take a censer full of burning coals of fire from off the altar before the LORD, and his hands full of sweet incense beaten small, and bring it within the vail: And he shall put the incense upon the fire before the LORD, that the cloud of the incense may cover the mercy seat that is upon the testimony, that he die not: And he shall take of the blood of the bullock, and sprinkle it with his finger upon the mercy seat eastward; and before the mercy seat shall he sprinkle of the blood with his finger seven times. Then shall he kill the goat of the sin offering, that is for the people, and bring his blood within the vail, and do with that blood as he did with the blood of the bullock, and sprinkle it upon the mercy seat, and before the mercy seat: And he shall make an atonement for the holy place, because of the uncleanness of the children of Israel, and because of their transgressions in all their sins: and so shall he do for the tabernacle of the congregation, that remaineth among them in the midst of their uncleanness. And there shall be no man in the tabernacle of the congregation when he goeth in to make an atonement in the holy place, until he come out, and have made an atonement for himself, and for his household, and for all the congregation of Israel. And he shall go out unto the altar that is before the LORD, and make an atonement for it; and shall take of the blood of the bullock, and of the blood of the goat, and put it upon the horns of the altar round about. And he shall sprinkle of the blood upon it with his finger seven times, and cleanse it, and hallow it from the uncleanness of the children of Israel. And when he hath made an end of reconciling the holy place, and the tabernacle of the congregation, and the altar, he shall bring the live goat: And Aaron shall lay both his hands upon the head of the live goat, and confess over him all the iniquities of the children of Israel, and all their transgressions in all their sins, putting them upon the head of the goat, and shall send him away by the hand of a fit man into the wilderness: And the goat shall bear upon him all their iniquities unto a land not inhabited: and he shall let go the goat in the wilderness. 
(Lev 16:7-22)

The two Birds: Lev. 14

And the LORD spake unto Moses, saying, This shall be the law of the leper in the day of his cleansing: He shall be brought unto the priest: And the priest shall go forth out of the camp; and the priest shall look, and, behold, if the plague of leprosy be healed in the leper; Then shall the priest command to take for him that is to be cleansed two birds alive and clean, and cedar wood, and scarlet, and hyssop: And the priest shall command that one of the birds be killed in an earthen vessel over running water: As for the living bird, he shall take it, and the cedar wood, and the scarlet, and the hyssop, and shall dip them and the living bird in the blood of the bird that was killed over the running water: And he shall sprinkle upon him that is to be cleansed from the leprosy seven times, and shall pronounce him clean, and shall let the living bird loose into the open field. And he that is to be cleansed shall wash his clothes, and shave off all his hair, and wash himself in water, that he may be clean: and after that he shall come into the camp, and shall tarry abroad out of his tent seven days. But it shall be on the seventh day, that he shall shave all his hair off his head and his beard and his eyebrows, even all his hair he shall shave off: and he shall wash his clothes, also he shall wash his flesh in water, and he shall be clean. And on the eighth day he shall take two he lambs without blemish, and one ewe lamb of the first year without blemish, and three tenth deals of fine flour for a meat offering, mingled with oil, and one log of oil. And the priest that maketh him clean shall present the man that is to be made clean, and those things, before the LORD, at the door of the tabernacle of the congregation: And the priest shall take one he lamb, and offer him for a trespass offering, and the log of oil, and wave them for a wave offering before the LORD: And he shall slay the lamb in the place where he shall kill the sin offering and the burnt offering, in the holy place: for as the sin offering is the priest's, so is the trespass offering: it is most holy: And the priest shall take some of the blood of the trespass offering, and the priest shall put it upon the tip of the right ear of him that is to be cleansed, and upon the thumb of his right hand, and upon the great toe of his right foot: And the priest shall take some of the log of oil, and pour it into the palm of his own left hand: And the priest shall dip his right finger in the oil that is in his left hand, and shall sprinkle of the oil with his finger seven times before the LORD: And of the rest of the oil that is in his hand shall the priest put upon the tip of the right ear of him that is to be cleansed, and upon the thumb of his right hand, and upon the great toe of his right foot, upon the blood of the trespass offering: And the remnant of the oil that is in the priest's hand he shall pour upon the head of him that is to be cleansed: and the priest shall make an atonement for him before the LORD. 
(Lev 14:1-18)

1. The Two Goats

Both goats deal with the sin question. The first goat (Christ) covers our sins (by His spilled blood).

And Aaron shall bring the goat upon which the LORD'S lot fell, and offer him for a sin offering. 
(Lev 16:9)

The second goat removes the sins (and we become righteous and inherit a new body where we no longer sin).

And the goat shall bear upon him all their iniquities unto a land not inhabited: and he shall let go the goat in the wilderness. 
(Lev 16:22)

2. The Wilderness

What and who is sent into the wilderness?

And Aaron shall lay both his hands upon the head of the live goat, and confess over him all the iniquities of the children of Israel, and all their transgressions in all their sins, putting them upon the head of the goat, and shall send him away by the hand of a fit man into the wilderness: 
(Lev 16:21)

And Jesus being full of the Holy Ghost returned from Jordan, and was led by the Spirit into the wilderness, 
(Luk 4:1)

3. Azazel

In ancient literature this name is similar to a fallen angel or diety - the devil. Azazel is derived from az, a female goat, and azel, to go away.

But is this goat Azazel representative of fall angel or diety, or *for* Azazel to use or tempt?

But the goat, on which the lot fell to be the scapegoat, shall be presented alive before the LORD, to make an atonement with him, and to let him go for a scapegoat into the wilderness. 
(Lev 16:10)

Then was Jesus led up of the Spirit into the wilderness to be tempted of the devil. 
(Mat 4:1)

4. Sin

*Your* study.

If you sin, are you or someone else responsible each time? Can someone else be held responsible or accountable for *all* of our sins? Christ? Azazel/Satan?

Who bore our sins?

And Aaron shall lay both his hands upon the head of the live goat, and confess over him all the iniquities of the children of Israel, and all their transgressions in all their sins, putting them upon the head of the goat, and shall send him away by the hand of a fit man into the wilderness: And the goat shall bear upon him all their iniquities unto a land not inhabited: and he shall let go the goat in the wilderness. 
(Lev 16:21-22)

Who his own self bare our sins in his own body on the tree, that we, being dead to sins, should live unto righteousness: by whose stripes ye were healed. 
(1Pe 2:24)

So Christ was once offered to bear the sins of many; and unto them that look for him shall he appear the second time without sin unto salvation. 
(Heb 9:28)

The next day John seeth Jesus coming unto him, and saith, Behold the Lamb of God, which taketh away the sin of the world. 
(Joh 1:29)

5. Being led into the Wilderness

And Aaron shall lay both his hands upon the head of the live goat, and confess over him all the iniquities of the children of Israel, and all their transgressions in all their sins, putting them upon the head of the goat, and shall send him away by the hand of a fit man into the wilderness: 
(Lev 16:21)

And Jesus being full of the Holy Ghost returned from Jordan, and was led by the Spirit into the wilderness, 
(Luk 4:1)

6. Leprosy

Leprosy, or Hansen’s Disease, is a chronic, crippling, progressive, skin disease which results in slow death, usually within ten to twenty years of contracting the disease. It has periods of debilitating weakness and fevers and is characterized by whitish patches of skin, tuberous skin growths, destroyed blood vessels, ligaments and skin tissues, scabs, ulcers, erosion of the bones, and the wasting away of tissues. It is very contagious during the periods of fever (Webster’s Dictionary, Zondervan Pictorial Encyclopedia of the Bible, vol. 2, pp. 138, 139, in part).

There is a cure for leprosy, today. We have no records to reveal what, if any, cure there was centuries ago. Leviticus 13, 14, 15 and 16 deal with leprosy, the clean and the unclean, and how one is to become clean.

For on that day shall the priest make an atonement for you, to cleanse you, that ye may be clean from all your sins before the LORD. 
(Lev 16:30)

This shall be the law of the leper in the day of his cleansing: He shall be brought unto the priest: And the priest shall go forth out of the camp; and the priest shall look, and, behold, if the plague of leprosy be healed in the leper; Then shall the priest command to take for him that is to be cleansed two birds alive and clean, and cedar wood, and scarlet, and hyssop: And the priest shall command that one of the birds be killed in an earthen vessel over running water: As for the living bird, he shall take it, and the cedar wood, and the scarlet, and the hyssop, and shall dip them and the living bird in the blood of the bird that was killed over the running water: And he shall sprinkle upon him that is to be cleansed from the leprosy seven times, and shall pronounce him clean, and shall let the living bird loose into the open field. And he that is to be cleansed shall wash his clothes, and shave off all his hair, and wash himself in water, that he may be clean: and after that he shall come into the camp, and shall tarry abroad out of his tent seven days. But it shall be on the seventh day, that he shall shave all his hair off his head and his beard and his eyebrows, even all his hair he shall shave off: and he shall wash his clothes, also he shall wash his flesh in water, and he shall be clean. And on the eighth day he shall take two he lambs without blemish, and one ewe lamb of the first year without blemish, and three tenth deals of fine flour for a meat offering, mingled with oil, and one log of oil. And the priest that maketh him clean shall present the man that is to be made clean, and those things, before the LORD, at the door of the tabernacle of the congregation: And the priest shall take one he lamb, and offer him for a trespass offering, and the log of oil, and wave them for a wave offering before the LORD: And he shall slay the lamb in the place where he shall kill the sin offering and the burnt offering, in the holy place: for as the sin offering is the priest's, so is the trespass offering: it is most holy: And the priest shall take some of the blood of the trespass offering, and the priest shall put it upon the tip of the right ear of him that is to be cleansed, and upon the thumb of his right hand, and upon the great toe of his right foot: And the priest shall take some of the log of oil, and pour it into the palm of his own left hand: And the priest shall dip his right finger in the oil that is in his left hand, and shall sprinkle of the oil with his finger seven times before the LORD: And of the rest of the oil that is in his hand shall the priest put upon the tip of the right ear of him that is to be cleansed, and upon the thumb of his right hand, and upon the great toe of his right foot, upon the blood of the trespass offering: And the remnant of the oil that is in the priest's hand he shall pour upon the head of him that is to be cleansed: and the priest shall make an atonement for him before the LORD. And the priest shall offer the sin offering, and make an atonement for him that is to be cleansed from his uncleanness; and afterward he shall kill the burnt offering: And the priest shall offer the burnt offering and the meat offering upon the altar: and the priest shall make an atonement for him, and he shall be clean. 
(Lev 14:2-20)

7. The First Bird/Dove

The issue of death is addressed - mortality and immortality. Two doves were needed to cleanse a leper. The first dove was killed in an earthen vessel over running water, and its blood and water and sprinkled (a type of baptism?) on the healed leper. The first bird is also representative of Christ since his blood is applied to us to.

And the priest shall command that one of the birds be killed in an earthen vessel over running water: 
(Lev 14:5)

Now when all the people were baptized, it came to pass, that Jesus also being baptized, and praying, the heaven was opened, And the Holy Ghost descended in a bodily shape like a dove upon him, and a voice came from heaven, which said, Thou art my beloved Son; in thee I am well pleased. 
(Luk 3:21-22)

8. The Second Bird/Dove

The second dove was dipped in the blood of the first dove, which ties the two works together. In the plan of salvation symbolized in the Torah by two doves, the first dove lays the foundation for immortality, but the second dove is needed for this (immortality) to occur by removing sin from us and making us righteousness.

As for the living bird, he shall take it, and the cedar wood, and the scarlet, and the hyssop, and shall dip them and the living bird in the blood of the bird that was killed over the running water: 
(Lev 14:6)

And I saw heaven opened, and behold a white horse; and he that sat upon him was called Faithful and True, and in righteousness he doth judge and make war. His eyes were as a flame of fire, and on his head were many crowns; and he had a name written, that no man knew, but he himself. And he was clothed with a vesture dipped in blood: and his name is called The Word of God. 
(Rev 19:11-13)

The next day John seeth Jesus coming unto him, and saith, Behold the Lamb of God, which taketh away the sin of the world. 
(Joh 1:29)

Note the root from which “baptize” comes from, and the analogy to the bird dipped in blood:

(G911. bapto¯? bap'-to A primary verb; to whelm, that is, cover wholly with a fluid; in the New Testament only in a qualified or specific sense, that is, (literally) to moisten (a part of one’s person), or (by implication) to stain (as with dye): - dip.)

9. The Open Field

The second bird was then released into an open field. (Christ comes into the world at His second coming).

And he shall sprinkle upon him that is to be cleansed from the leprosy seven times, and shall pronounce him clean, and shall let the living bird loose into the open field. 
(Lev 14:7)

The field is the world; the good seed are the children of the kingdom; but the tares are the children of the wicked one; 
(Mat 13:38)

10. The Fall Festivals represented?

But it shall be on the seventh day, that he shall shave all his hair off his head and his beard and his eyebrows, even all his hair he shall shave off: and he shall wash his clothes, also he shall wash his flesh in water, and he shall be clean. 
(Lev 14:9)

In the last day, that great day of the feast, Jesus stood and cried, saying, If any man thirst, let him come unto me, and drink. He that believeth on me, as the scripture hath said, out of his belly shall flow rivers of living water. (But this spake he of the Spirit, which they that believe on him should receive: for the Holy Ghost was not yet given; because that Jesus was not yet glorified.) 
(Joh 7:37-39)

Is the last great day that of the Feast of Tabernacles (not the 8th Day), which is why Christ refers to rivers of living water?

Each day of the Feast of Tabernacles in Jerusalem, a celebration called Simkhat Beit Hashoavah ("The Rejoicing at the Place of the Water-Drawing.") takes place. The Levitical priests went to the pool of Siloah (Silwan) to fill a golden flask and were greeted by shofar blasts at the Temple's Water Gate. They then poured the water so that it flowed over the altar together with wine from another bowl.

"For though that ceremony was considered by the Rabbis as being a subordinate reference to the dispensation of the rain, the annual fall of which they imagined was determined by God at that feast, its main and real application was to the future outpouring of the Holy Spirit, as predicted -probably in allusion to this very rite -- by Isaiah the prophet. Thus the Talmud says distinctly: 'Why is the name of it called, The drawing out of water? Because of the pouring out of the Holy Spirit, according to what is said: 'With joy shall ye draw waters out of the well of salvation."' Hence, also, the feast and the peculiar joyousness of it, are alike designated as those of 'the drawing out of water'; for according to the same Rabbinical authorities, the Holy Spirit dwells in man only through joy" (The Temple, by Alfred Edersheim, p.279-280).

And on the eighth day he shall take two he lambs without blemish, and one ewe lamb of the first year without blemish, and three tenth deals of fine flour for a meat offering, mingled with oil, and one log of oil. And the priest that maketh him clean shall present the man that is to be made clean, and those things, before the LORD, at the door of the tabernacle of the congregation: 
(Lev 14:10-11)