What are the differences between a claim, a statement and a proposition?

In contemporary mathematical logic, we generally ignore some philosophical distinctions. My guess is that you do not mean "proposition" here as the label that sits beside some theorems in a book, and instead you mean it in a philosophical sense.

In this sense, a statement expresses a proposition, but the same proposition could be expressed by multiple statements. A theorem is then a statement that has been proved. Or is it a proposition that has been proved? You can see that there are many complications here.

The existence, or lack thereof, of abstract "propositions", independent of the sentences that express them, is an important topic in philosophy. So you may have encountered a book that takes a more philosophical approach and mentions the distinction. But, especially at first, you can likely just identify propositions with statements, and everything will be fine.

Here's my take on this. A statement is indeed a sentence which can be true or false. A proposition is a statement that the author is proposing for further scrutiny, possibly a proof. A claim is a proposition that the author claims is true. The differences are merely subtle characterizations by the author -- all are statements.

Prior to the edit, you mentioned theorem, so I'll elaborate further.

A theorem is a statement (including a proposition or claim) that has been proven true (or sometimes one that is very soon to be proven true). A corollary is a theorem that follows in a obvious or simple way from another theorem. A lemma is a theorem that is very useful in the proof of another theorem or theorems. Again, the differences are characterizations by the author -- all are theorems.