What is the relationship between the Abrahamic Covenant and the Mosaic Covenant? Also, what is the relationship between the Abrahamic Covenant and the New Covenant? Are Christians still under the Abrahamic Covenant?
This is a good question. I believe that it is (more or less) answered in Galatians 3:10-22. In this passage Paul tells us that those of us who are saved through faith in Jesus Christ are children of Abraham. This is also reiterated in Galatians 4:21-31. Paul deals with the relationship between the Law of Moses and the covenant with Abraham in 3:15-20. He tells us that the law, delivered 430 years after the covenant with Abraham, does not nullify the covenant with Abraham. “The law, introduced 430 years later, does not set aside the covenant previously established by God and this do away with the promise.” In fact, Paul argues that the covenant with Abraham, which is that salvation is by faith, supersedes the Law of Moses. In Christ, the Law (not just the Law of Moses, but law in general) was nullified at the cross (Colossians 2:13-15).
We do not normally describe ourselves as Christians as being under the covenant of Abraham. We describe ourselves as being under the New Covenant. Hebrews 8 talks about why the New Covenant is superior to the Old Covenant, which is the covenant with Moses. The Hebrew writer also uses Jeremiah 31 to make his point. So, it is not our habit to describe ourselves as being under the covenant with Abraham, but that, in essence, is what Paul says in Galatians 3. The covenant with Abraham is a foreshadow of the covenant we have in Christ. Abraham is the father of us all (Romans 4:16).
The covenant with Abraham concerned his physical descendants, which are the Jews, but it also concerned all humanity. Genesis 12 and 15 tells us that all nations would be blessed through Abraham. This was not true of Moses. One can argue that the covenant with Abraham is greater and more significant than the covenant with Moses. However, Paul may not have put it that way, as he goes to great lengths to appreciate the blessings of the Law of Moses in Romans 9:1-5 and Romans 11:1-6. So… I do not feel it is productive to say that the covenant with Abraham is “better” than that with Moses. Each served its purpose in the overall plan of God. Without the Law of Moses, there would not have been Israel, and the Jewish prophets and most of the Old Testament. But, there is a sense in which the covenant with Abraham is a more general and lasting covenant than that with Moses.