Keeping with the free theme, Book Of Proof by Richard Hammack is a nice introduction to proofs and formalism. It's available free from the author as a PDF[1], and also as a physical book on Amazon[2].

An alternative if you're willing to spend a little is How to Prove It by Daniel J. Velleman, also available from Amazon[3] and probably many other retailers. Both books cover roughly the same topics.

[1]: http://www.people.vcu.edu/~rhammack/BookOfProof/

[2]: http://www.amazon.com/Book-Proof-Richard-Hammack/dp/09894721...

[3]: http://www.amazon.com/How-Prove-It-Structured-Approach/dp/05...

This stuff is usually rolled into courses called "abstract algebra". If a textbook is called "abstract algebra", it's usually designed for a first year undergraduate. If it's just called "algebra", it's usually aimed at a more mature audience. Herstein and Hungerford are the texts I learned from. This book seems more recent and popular:

http://www.amazon.com/Book-Proof-Richard-Hammack/dp/09894721...

The other big stream of basic undergraduate mathematics is analysis. For that I recommend Spivak's Calculus.

Coursera's Calculus One and Calculus Two courses.

I've found other branches of math to be much easier to understand if you have basic knowledge about calculus.

I really like Hubbard & Hubbard.