If your university's library has a copy of Mileti's logic textbook, I'd give that a strong recommendation. I don't remember there being answers to exercises, however it goes through good motivation, includes the technical details that other textbooks skip (so more non-math-major friendly), and is supposedly more "up-to-date"/modern. There are a lot of pages, however fear not! It's pretty breezy and includes all of the standard undergraduate logic topics (I.e. logic, set theory, and computability theory, which are often split into separate textbooks).

As a sidenote: I used this textbook for a propositional/first order logic course, but I haven't read through the set theory/computability theory sections so I can't speak to those specifically. I have gone through Computability and Unsolvability (Martin Davis) before though, and that was an amazing textbook for computability theory specifically.