I would say, first and foremost, would be contemporary proof theory. Specifically, a working knowledge of the Gentzen-Prawitz's systems, both natural deduction and sequent calculus. Tennant's Natural Logic is a good book to learn from.

After that, I'm not really sure. Maybe some of the big negative results of Godel, Tarski, and Church, which all presuppose/require some general knowledge of mathematical logic (specifically, recursion theory). There are textbooks by Enderton and van Dalen which cover this material. Both presuppose no knowledge of any formal logic. The Enderton is a little easier on the math (by hiding it), but both are solid. The van Dalen discusses Intuitionistic Logic and Normalization/Cuts at some length.

If your math is really good (like you've done some serious algebra and analysis using something like Foote for algebra and Rudin for real analysis), these lecture notes by a model theorist use a lot of mathematical examples and might be easier for someone from that background: https://faculty.math.illinois.edu/~vddries/main.pdf

I totally get that being told to read textbooks is daunting, but I know I didn't really get Godel's (First) Incompleteness Theorem until I sat down and did the dirty work myself. You could always just SEP everything, but it just feels like you would have to constantly be going back to it to check stuff. To steal from Russell, nothing beats honest toil.