If you are looking for a general grounding in Western philosophy, look up the books in a university philosophy curriculum. If you are looking for books that are likely to interest someone with your background, I have some suggestions. You probably want to focus on analytic philosophy, mostly from the English-speaking world and, at least initially, avoid classical philosophy from Greece, and continental philosophy from France and Germany (with a few exceptions such as Frege, Descartes, Kant, and Duhem).
I'd start with
which is often viewed as a foundational work on analytic philosophy. Other great works on the foundations of mathematics include
On metaphysics and epistemology, I'd recommend
Kant is by far the most difficult of the metaphysics books I'm recommending, and it would be worth your while to read some commentaries as well. There is lots of great material on-line.
On the philosophy of science, I'd recommend Hume's Enquiry as a critical text, before reading anything that follows it historically, and the following:
You'll notice almost all of my recommendations are from before 1950, but I think these authors are necessary for a strong grounding in understanding what more recent philosophy is about. Read some of the books I recommend, decide which ones interest you the most, and then look for discussions and commentary on those, which will direct you to more modern work.
A few comments on what I've left out: a lot of people think a grounding in classical philosophy is important, but Galileo, Descartes, Newton, and Locke pretty much put an end to classical philosophy, and I don't see it's influence much in anything that comes after those authors. Francis Drake gets a lot of attention as the "first modern empiricist", but I found his work unimpressive. Wittgenstein was very influential, but more in the sense of scattered ideas than in any systematic sense, and I don't know that he's really worth reading unless you are curious about it. If you think Kant is impressive, you might think that it would be interesting to read the German idealists who followed Kant, but you would be mistaken. Most of them are typical continental philosophers, meaning that they are quite opaque to most English speakers who think analytically.