As I'm a bit further on in this path, so the details of the beginnings (introductions to proofs and logic) are a bit blurred to me, but I'll present a few books that I think are good for moving from basics to graduate level topics.
In number theory to move from basics to graduate level topics would be Hatcher's Topology of Numbers into Silverman and Tate into Koblitz.
For linear algebra I'd recommend Axler's book, for complex analysis I liked Churchill and Brown in that I could basically read through and do the exercises very quickly as an undergrad. I didn't have a great time with any of my real analysis texts so I don't think I can give a good recommendation for that.
Of course your mileage may vary but these are the books that I've enjoyed the most, in that they stay grounded in computations, give decent intuition with good geometric pictures, and tend to have approachable exercises compared with other books I've read.
Also: learn to program in python, and use SAGE to solve project Euler problems.
There is a great 1st real analysis book that would work from HS level: "S. Abbott (2001). Understanding Analysis. Undergraduate Texts in Mathematics." (For comparison, Baby Rudin would be way more advanced than that, I'd schedule it even after Axler's Linear Algebra, which itself should go after a more matrix-y introduction to linear algebra, like Strang.)
This will not be a long post; I have a simple question to ask: if you wanted to educate yourself to graduate level in mathematics, but didn't actually want to go to university, what would you do? I would ask for text-book recommendations, but I don't want to limit your responses (however, bear in mind that the wikipedia articles on, say, cardinality or well-ordering go over my head – they may skim my hairline, but over they go). Also bear in mind that while I personally have A-levels (British qualifications) in both Maths and Further Maths (which is to say, I know some calculus at least), there are probably plenty of people on lesswrong who don't and who desire the same information – so assume as much ignorance as you feel necessary (it's a shame, actually, that there isn't a sequence here on lesswrong for maths). What do you advise (if you think the query ill-defined, I would like to know that as well)?