Is there a book on proofs with solutions? I am a biochemistry graduate student who works on cancer. I am interested in learning proofs as a personal interest. I use math as a tool, but would like to start building a deeper understanding on my own. I am not taking any course. Hence, I am looking for a book with theory, exercise, and solution manual, in case I am stuck. I find this forum extremely helpful, but I would still like to have a reference. Most books that I have started looking to buy do not have a solution manual. Can anyone recommend an author? Sorry for this general question. Thank you! 

EDIT:

I watched the movie Good Will Hunting, so I feel confident! lol I
In my experience you won't tend to find a book with a solution manual and if you do you won't find one that does a lot of exercises as proofs can get pretty long and tedious in a hurry. Most authors, I think, assume that you will be interacting a fair amount with your professors/other students at that point so I don't think they get written very much.

For a basic text into abstract math I would suggest my undergraduate text "Bridge to Abstract Mathematics" it does a good job of making important remarks and breaking things down. There is, no surprise, not a solution manual that I am aware of but I have a set of YouTube videos going up that go through and lecture on each chapter and do a number of the examples/problems in detail. They will start going up in a few weeks under the username superphyz and I will link them in the comments later if you comment and let me me know if that will be useful to you. 

Good luck!
Well, there are lots and lots of books with solutions.

An answer to your question depends on which part of mathematics you would like to explore. For instance, a series of great ones about real analysis come from Kaczor &amp; Nowak