I wanted to ask you a question about the testaments. Is the Old testament in particular the sacrifice of the animals, and the new testament is the shed blood of Jesus for forgiveness? Am I correct?

A testament is a covenant that is based on something received as the result of someone or something dying (Hebrews 9:16-17 [16] For where a testament is, there must also of necessity be the death of the testator. [17] For a testament is of force after men are dead: otherwise it is of no strength at all while the testator liveth.
See All...). We still use the word today in the idea of a last will and testament--a document which determines who gets what after the author of the will and testament dies.

The "old testament" technically refers to the covenant of the law made with the children of Israel. In 2 Corinthians 3:14But their minds were blinded: for until this day remaineth the same vail untaken away in the reading of the old testament; which vail is done away in Christ.
See All..., the "old testament" refers specifically to the written law of Moses--"But their minds were blinded: for until this day remaineth the same vail untaken away in the reading of the old testament; which vail is done away in Christ." The "old testament" was read and therefore refers to the writing here.

However, the old testament is also called the "first testament." Hebrews 9:15And for this cause he is the mediator of the new testament, that by means of death, for the redemption of the transgressions that were under the first testament, they which are called might receive the promise of eternal inheritance.
See All... speaks of "the redemption of the transgressions that were under the first testament." This indicates that the first, or old, testament did not completely do away with sins but simply held them in check until the perfect sacrifice of Christ. But the first testament did require the shedding of blood. Consider this passage:

Hebrews 9:18Whereupon neither the first testament was dedicated without blood.
See All... Whereupon neither the first testament was dedicated without blood. 19 For when Moses had spoken every precept to all the people according to the law, he took the blood of calves and of goats, with water, and scarlet wool, and hyssop, and sprinkled both the book, and all the people, 20 Saying, This is the blood of the testament which God hath enjoined unto you. 21 Moreover he sprinkled with blood both the tabernacle, and all the vessels of the ministry. 22 And almost all things are by the law purged with blood; and without shedding of blood is no remission.

This passage refers specifically back to an incident when Moses sprinkled the blood of the covenant in Exodus after the people agreed to keep the commandments of the law.

Exodus 24:7And he took the book of the covenant, and read in the audience of the people: and they said, All that the LORD hath said will we do, and be obedient.
See All... And he took the book of the covenant, and read in the audience of the people: and they said, All that the LORD hath said will we do, and be obedient. 8 And Moses took the blood, and sprinkled it on the people, and said, Behold the blood of the covenant, which the LORD hath made with you concerning all these words.

As the old testament was based on the sacrifice of animals (Hebrews 9:18Whereupon neither the first testament was dedicated without blood.
See All...), so the new testament is based on the sacrificial death of Jesus Christ. Jesus is called "the mediator of the new testament" (Hebrews 9:15And for this cause he is the mediator of the new testament, that by means of death, for the redemption of the transgressions that were under the first testament, they which are called might receive the promise of eternal inheritance.
See All...). That is why He called His shed blood (as pictured in the fruit of the vine) "my blood of the new testament" (Matthew 26:28For this is my blood of the new testament, which is shed for many for the remission of sins.
See All...). The new testament is the source of eternal life for those who believe in Jesus Christ. In contrast to the old testament, the new testament provides full forgiveness for sins and eternal salvation.

Hebrews 10:10By the which will we are sanctified through the offering of the body of Jesus Christ once for all.
See All... By the which will we are sanctified through the offering of the body of Jesus Christ once for all. 11 And every priest standeth daily ministering and offering oftentimes the same sacrifices, which can never take away sins: 12 But this man, after he had offered one sacrifice for sins for ever, sat down on the right hand of God;

In time, the inspired books that were written before the coming of Christ came to be known as the Old Testament, while the inspired books that were written after Christ came and died on the cross were known as the New Testament. However, the phrases originally referred to two foundations for forgiveness--the sacrifice of animals and the sacrifice of Jesus Christ.