I'm somewhere a long this path, so I can share my experiences. Something worth playing with is Coq, I enjoyed doing the most of course here. One benefit is that it uses a consistent notation to represent things and not too many unfamiliar symbols. It introduces things like the Peano axioms and other interesting stuff. It will also nail down commutativity, transitivity and other useful concepts. Not sure how useful it is, if you haven't done some programming.

For learning important symbols and terminology, I would look at Set theory. Knowing the difference between "member of" symbol and "subset of" symbol of is fairly basic stuff, I just picked it up by osmosis. Predicate calculus also crops up a fair bit, I got taught it in my computing degree. I've got Conceptual Mathematics because I found a fair amount of the maths around functional programming languages was expressed in terms on Category Theory. It was good for the first few chapters but my interest waned at some point and I find it hard to get back into,

I'm interested what other people would recommend.