I'm not a big fan of full roadmaps and reading lists. Exploring mathematics is something that can be totally different depending on where and who you are. Any serious roadmap needs to be flexible and take account of the course: reading maths is a skill (one you seem well on your way to learning btw! but still...), initially you may find actual teaching easier to grasp- and your reading should work along side that. So here's my attempt at a flexible roadmap:
1) Buy some very carefully chosen books and read them cover to cover: There's a lot of baffling books out there- even some that look really UG friendly can have you weeping by page 5 in your first year- and you only want 4-5 to start with (any more will just be too expensive and you won't get round to reading all of them- top up via the library). My recommendations are: 'Naive set theory- Halmos', 'Finite dimensional vector spaces- Halmos', 'Principles of mathematical analysis- Rudin' and 'Proofs from the book- Aigner and Ziegler'.
[These, ostensibly, cover the exact same material as the book you have decided to buy- but to develop quickly I urge you to buy more mathematical texts like these: Halmos' and Rudin's writing styles are very clear but technical (in a way that the book you are interested in will not be), and will make you a better mathematician faster than any book that tries to 'bridge the gap' ever would. I also seconded Owen's call for proofs from the book: it is simultaneously inspiring and useful as a way of seeing 'advanced' topics in action- it's something you'll keep coming back to, right up to your third year!]
2) Do all of the excercises: Or as much as you can bear to- even if it looks like it's beneath you (if you're half decent- a lot of first year will!) you will be surprised as to how much it helps with your mathematical development (and the crucial high mark you'll need for a good PhD placement). This applies to classes and your 4-5 text books.
3) Ask your tutor about doing some modules from the year above: If you've read all of those textbooks and done all of the excercises, you will be ready. Get some advice from your tutor about what would be best and roll with it (most unis won't make you take the exam, so if you don't feel comfortable you're fine). Taking something like metric spaces or group theory in your first year will put you top of the pile.
4) Keep doing all of these things: Immerse yourself in maths- keep on MO, meet likeminded people and no matter how slow the course seems to be moving, no matter the allure of apathy: keep at it. Advice for later books would be pointless now, but there will be people who can give it to you there and then (use maths forums if you want). Oh, and never rule out an area- you never know where intrigue will come from...
Best of luck,
Tom
For set theory, I recommend Drake's "Set theory: An introduction to large cardinals" (which also contains a good treatment of various other parts of set theory, before concentrating on large cardinals) and Kunen's "Set theory: An introduction to independence proofs." The "bible" of set theory is Jech's "Set Theory" but I think it may be easier to learn from Drake and Kunen first. [Do you get the impression that set theorists are not very creative when it comes to book titles?]
For logic in general, Shoenfield's "Mathematical Logic" covers an amazing amount of material in rather few pages. Most of it is great (if you do the exercises, into which a lot of material has been squeezed) but I wouldn't recommend his approach to constructibility and forcing --- use Kunen instead.
Best Answer
Pólya-Szegő seems unsurpassed as a graduate level problem book on classical function theory. Other classic examples are:
P. Halmos, A Hilbert space problem book,
A. Kirillov and A. Gvishiani, Theorems and problems in functional analysis, available in English and French, bseides the Russian original, and
I. Glazman and Yu. Lyubich, Linear analysis in finite-dimensional spaces, translated from the Russian.