Since you have a background in computer science you might appreciate some of the following books:
- Formal Logic by A. N. Prior
- Elements of Mathematical Logic by J. Lukasiewicz
- Aristotle's Syllogistic: From the Standpoint of Modern Logic by J. Lukasiewicz
- Polish Logic 1920-1939 Edited by Stors McCall
Prior's book has sections on propositional calculus, quantification theory, the Aristotelian syllogistic, traditional logic, modal logic, three-valued logic, and the logic of extension. The section on propositional calculus includes a probably not very well known topic in a short of exploration of what can get variable functors. or equivalently variable truth-functions (meaning that we have at least one truth-function which qualifies as a logical variable instead of a logical constant).
Elements of Mathematical Logic starts out with a discussion of the history of logic, and then proceeds to develop two-valued propositional calculus axiomatically, using what I believe a very simple axiom set. The axioms in words can get stated:
1) "If the first (proposition) implies the second, if the second implies the third also, then the first implies the third."
2) "If the negation of the first implies the first, then the first."
3) "If the first, then if the negation of the first, then the second."
Other sections include two-valued propositional calculus with quantifiers, problems of the independence of the axioms presented, consistency of the system, and completeness of the system. The book has much to recommend it, in my opinion, for learning the axiomatic method, as well as realizing that many, many relationships exist or can get formed between logical laws and metatheoretical results. Wajsberg's papers in the Polish Logic volume makes this even more apparent.