Truth Trees
A fun way to prove propositional logic statements. More information on truth trees
here, here, here, and here. See my own study sheet here.

Truth Tree Solver
If you can't wait to do your proof by hand, this solver will generate a proof tree for you. Enter the proposition to be tested in the input field and out comes the completed tree.

Formal Logic Primer
Accessible yet rigorous formal logic text produced for students at U.C. Davis. Covers sentence and predicate logic. Two volumes, plus exercises
and solutions manual!

Mathematical Logic (Simpson)
Lecture notes for intro courses in mathematical logic offered at The Pennsylvania State University. A more traditional, formal logic text. Good companion to primer above.

How to Read and Do Proofs (Solow, 3rd)
Classic work on proof writing. I own all five editions – including the first, which I used while taking Abstract Algebra in college. 3rd edition is best. Short, sweet, and indispensable.

Book of Proof (Hammack, 3rd)
Easy-to-follow formal textbook covering basic logic and proof writing. Excellent adjunct to the other books referenced on this page. Like the others, it's presented as a free pdf.

Introduction to Mathematical Thinking
Stanford Online course by Keith Devlin, renowned professor and math education guru. Devlin is famous for presenting difficult material in an unintimidating way to lay audiences.

Gödel, Escher, Bach (Hofstadter)
Mind-bending tome about self-reference in art, music, math, and meta-logic. Inverted melodies. Up is down. This sentence is false. A tantalizing, formidable intellectual challenge.