Why did God create a new covenant? It seems as if he had a covenant set in place in the Old Testament and provided a means for Israel to follow the covenant and offer sacrifices to atone for their sins. Why did he create a new covenant in the New Testament which is the basis of Christianity? Did mankind or God change in a way that a new covenant needed to be set up?
If God had created Christianity first, people would not have known their need for a Saviour. "What do we need to be saved from?
So God first sets up a do-it-yourself religion; by which I mean that if the Jews followed all the laws they could save themselves earning their right to heaven.
That didn't work. Worse, some people became more interested in the laws than the purpose of the laws:
See also: a brief summary of how salvation works.
But now people could at least compare their standards to God's and realise the gap between themselves and God:
If you want to read more about this, the book of Hebrews explains more about how Judaism (and particularly the law) was a shadow of the forthcoming revelation. For example, the animal sacrifices in Judaism teach us that a sacrifice is required for the forgiveness of sins. This is how we can understand passages like this:
You've actually got it inside out. There's a lot of evidence, both from biblical and extrabiblical sources, that the basic ideas of Christianity were around from the beginning. (See When did Christianity originate? for just a few examples.) The "new" covenant was the Law of Moses, which was given to the people of Israel--at the time a rowdy group of recently-freed slaves--after they demonstrated that Moses literally could not leave them alone for long enough to go up Mt. Sinai, get some commandments from God and come back down without them falling into degenerate heathen fertility-worship.
These were people with no chance of successfully living the high principles of the Gospel, so God gave them a lesser law that focused strongly on physical, tangible things, as a "schoolmaster" to point their minds towards Christ and his atoning sacrifice. (Galatians 3:24-25) Christ fulfilled the Law, and it was no longer needed, so they went back to the original covenant and the original Law of the Gospel. (And by this time the observation of the Law of Moses had been twisted by Jewish leaders to the point where it was a lot closer to "worship of the Law" than "worship of the Lord" anyway.)
God made several covenants throughout time with different people or groups of people, including Adam, Noah, and Abraham.
So, after the initial covenant with Adam, we could ask why a new covenant was made. The answer is that the first covenant was broken by Adam. With Noah, it wasn't that a previous covenant was broken, but that God made it clear that He would never destroy the earth with water again.
The covenant with Moses and the nation of Israel was in effect for around 1500 years, from the time of Moses to the time of Christ. Now it could be argued that it is still in effect with the nation of Israel, I know.
This covenant involved the obedience of the people of Israel to the Mosaic Law and God's blessing or curse based on that obedience. However, the covenant never achieved righteousness in the blood of bulls and goats (as Hebrews states).
During this covenant, an new covenant was spoken of. The New Covenant was initiated with the resurrection of Jesus and replaces the old covenant. This is a better Covenant in that it does impute righteousness to mankind and brings forgiveness and redemption. The old covenant was put in place as a tutor to bring us to Christ.
So, God created several "new" covenants throughout history for various reasons. The newest of them is the New Covenant in the shed blood of Jesus, as He indicated at the last supper. It's a better covenant in that it is eternal and in that is brings redemption to mankind.
Jesus was Judaism fulfilled. Judaism made perfect. The prophet Daniel foretold the coming of the messiah and in Christ his prophecy was fulfilled.