Answer to A. Barwise & Etchemendy's Language, Proof, and Logic is a decent introduction to logic. From the perspective of someone coming from a CS background I found it especially accessible and fun. It's very clearly written, has lots of examples, and comes with 3 programs:
- Tarski's World: is a block world where you can verify the truths of sentences or find counterexamples in an intuitive, visual environment
- Fitch: is a Fitch-style proof system that will check your proofs
- Boole: is a truth-table tool
Once you're done with the main course (I-II), in part III they have more fun topics to explore:
- program correctness
- metalogic (e.g. Lowenheim-Skolem, Compactness, Incompleteness)
As I said, B & E is an introductory course, so next you can move to what's called mathematical logic, which is more technical but more precise and will turn out to be more rewarding in the long run. A canonical text here is Enderton's A Mathematical Introduction to Logic, which even has a special chapter dedicated to Second-Order Logic (something you said you're interested in). It's not as interactive as B & E, but it's a classic and it will prepare you for more advanced logic.
Answer to B. How to give the semantics of operators should become clear after studying these texts. But if you're looking for special kinds of semantics, such as Kripke, topological, neighborhood, measure-theoretic, domain-theoretic, probabilistic, etc., you'll have to do a little digging to find papers that explain how these things are done.
Hope that addresses at least some of your questions.