How do different Protestant denominations view the relationship between Israel and the Church? I know there is some complicated theology, which I am not an expert in, regarding what I am about to ask.  I honestly don't even know the technical terms for what I am trying to describe.  I would like an overview of how different Protestant Christians understand this relationship, explaining the different bases for the following possibilities:


The belief that Israel "became" the Church
The belief that the Israel is still important and hasn't been entirely replaced by the Church
It sounds like you are wondering about the debate between covenant theology and dispensationalism.  These are big questions, and only a very high-level summary can be provided here.

Covenant theology holds that there is one people of God, and that the saved from both Israel and the Christian church are members of that people.  Thus, God's promises to Israel in the Old Testament apply to the Christian church as well.

Dispensationalism sees Israel and the Christian church as distinct groups, and that the promises to Israel in the Old Testament generally do not apply to the Christian church.

Take note too that there are many other categories besides these two frameworks.  Ideas like dual-covenant theology and supersessionism ("replacement theology") are sometimes seen to be more extreme forms of dispensationalism and covenant theology, while frameworks like New Covenant Theology are seen to be somewhere in the middle.

Regarding denominations, remember that there are thousands of Protestant denominations, so the best we can do on a site like this is a high-level summary.  That said, Reformed and Presbyterian denominations tend to be adherents of covenant theology.  Baptist denominations and especially "non-denominational" churches tend to be dispensational.  But most denominations do not take an official stance, or their stance is general enough that it allows for friendly disagreement within their churches.