Reddit reviews Discrete Mathematics and Its Applications, Global Edition

Wow. That's quite a range - normally those would be covered in two, three, or even four courses. Any wonder the pass rate is so low.

It's difficult to recommend anything sensible for the whole course - I'd have suggested Rosen - I have found it somewhat useful as a reference text later on - but it seems to be unpopular with amazon reviewers. For the latter part (finite-state machines etc), Sipser is a common course textbook - but it may be too advanced for a starting point (though might be worth getting so you can get ahead before you get to that part of the course, which will almost certainly be the trickiest).