On my day job I write software for Google. I am broadly interested in theoretical computer science and psychology. Recently I've been thinking more about existential risks and risks from advanced AI in particular.
Ahh that's because in India we have a graduate entrance exam called GATE (for admission to IITs and other institutes). And ToC carries a lot of weight-age.
I found Introduction to Automata theory by Ullman on EdX to be extremely useful. It covers most of the basics and there's also a book by the same author. Apart from the lectures you also get access to high quality assignments, problem sets which I think are very essential for a course like ToC.
Yeah even I used to feel the same, Wigner wrote an article about it "The unreasonable Effectiveness of Mathematics in the Natural Sciences" you will definitely find it useful.