You can take four terms of calculus using the Stewart book and not need to do any proofs. You do need to review algebra and have a good grounding in that, though.

You can also cover linear algebra without proofs.

There are, of course, books that do cover proofs for calculus and linear algebra, but it depends on what you’re trying to get out of the material.

I have heard that proofs are a big part of advanced math, so that is one reason I am interested in proofs.

I should have listed proofs as a third topic.

You're right it is a big part of advanced math, but you can cover calculus, linear algebra, and proofs in parallel! If you have enough time.