Here are a few introductory books that I've found useful. I'm in a somewhat similar situation as you, I suppose--a year older, but having had to learn whatever mathematics that I know mostly independently (though I've been lucky to have been able to talk to several mathematicians on various occasions as well as take a couple of courses). For more advanced ones, I'd certainly be the wrong person to ask. Anyway, here are some of the books that I've used (or am currently using).
Algebra: I think Herstein's Topics in Algebra is often recommended as a good introductory-level undergraduate textbook (which I learned material from). Lang's Algebra is a nice follow-up, but I think a lot of people (certainly including myself) would be turned away from the subject if Lang were the first thing they opened.
Analysis: Rudin's Principles of Mathematical Analysis is great, but the real fun is in Real and Complex Analysis (you don't have to read all of the former to get into the latter). Also, Peter Lax's Functional Analysis is very enjoyable reading.
Differential geometry: I personally found it very hard to read Spivak's Calculus on Manifolds because there wasn't much motivation (like, why would one define a tangent space when you're just working with open subsets of $\mathbb{R}^n$?), but volume 1 of his A Comprehensive Introduction to Differential Geometry has similar material with much more explanation and motivation.
Number theory: My first introduction to number theory was from Niven and Zuckerman's book. Serre's A Course in Arithmetic is also very interesting, but I would consider that more of a second or third read (I tend to find Serre's books rather terse and consequently struggle with them, but this is likely just a personal failing). For instance, you need to know what a projective limit is to read it.
I wish I knew a really good introduction to algebraic number theory. The ones I've seen have tended to be somewhat difficult (i.e., presupposing a fair bit of material in Lang's algebra such as familiarity with localization, noetherian rings, etc.--topics that might be omitted in a first course on abstract algebra). I think Lang's Algebraic Number Theory is a great book, but it took me an enormous amount of time before any of it started to make any sense at all. And there are other "elementary" books on algebraic number theory that never get anywhere interesting.
Linear algebra: I learned this from Hoffman and Kunze's book, but I think the subject is nicer in a more abstract context (e.g. after one has talked about rings).
Computer science: Wait, hold on! I think I'd be remiss if I didn't mention Sipser's An Introduction to the Theory of Computation, which has to be one of the most enjoyable books I've ever seen---and it is basically mathematics.
Logic: Ebbinghaus, Frum, and Thomas have a very nice book on mathematical logic in the UTM series.
Topology: Dugundji's book is a bit old, but I found the exposition very crisp and enjoyable. At the same time, I suppose it's not good for geometric intuition. I have heard great things about the books by Munkres but have not (yet) read them myself.
The books in the Carus mathematical monograph series are all accessible, pithy, and enjoyable; Krantz's Complex Analysis: The Geometric Viewpoint is one that I'm enjoying looking at right now.
Another bit of advice: You don't have to finish a book to "graduate" to another one! Skipping around is something many mathematicians do, and one never "really" understands an area of mathematics (at least, not as an undergraduate), so it's more efficient to move on to other things. Nor do you have to know everything. (This isn't me giving advice; it's me regurgitating something that a very respected mathematician told me a while back, and was quite a revelation for me.)
I obviously don't know whether you're near a university library and have borrowing privileges; I happen to live near three (albeit ones from liberal arts schools with no graduate math department). If not, I strongly recommend buying used books, since math textbooks tend to be ridiculously overpriced for some reason. Fortunately, there are many good resources on the web: James Milne's site is excellent, for instance.
Anyway, if you need more, the bibliography on my blog has a longer list (but the ones here should probably keep you busy for at least a little while). You could also try contacting a professor at a nearby university to see if he or she is willing to mentor you for a research project; there may be programs through which this is possible (though admittedly I have no idea how it works in Europe). The benefit of this is that you'll end up absorbing a lot of new mathematics along the way as well as better understanding what you already know.
