[fmap]

What exactly are you trying to learn? Mathematical logic is a huge field in its own right, with plenty of topics that are of historical interest and a lot of active research areas.

If you want to learn modern mathematical logic you're in for a rough time, since you'll basically have to learn category theory in order to understand the few really excellent textbooks which exist (e.g. Sketches of An Elephant). If you are interested in type theory you should try reading the Homotopy Type Theory book, which is (mostly) self contained.

[imakecomments]

I'm mostly interested in "modern" mathematical logic then. I'm very interested in learning category theory and its connections to programming language theory. I already know a little bit of category theory, but am open to any good beginner sources. I'm also interested in classic recursion theory and a bit of proof theory with its connections to CS. I don't know many people doing any of this and it doesn't seem that popular in math departments.

[wolfgke]

Category theory is not logic.