I think people generally agree that analysis, topology, and abstract algebra together provide a pretty solid foundation for graduate study. (Lots of interesting stuff that's accessible to undergraduates doesn't easily fall under any of these headings, e.g. combinatorics, but having a foundation in these headings will equip you to learn those things quickly.)

For analysis the standard recommendation is baby Rudin, which I find dry, but it has good exercises and it's a good filter: it'll be hard to do well in, say, math grad school if you can't get through Rudin.

For point-set topology the standard recommendation is Munkres, which I generally like. The problem I have with Munkres is that it doesn't really explain why the axioms of a topological space are what they are and not something else; if you want to know the answer to this question you should read Vickers. Go through Munkres after going through Rudin.

I don't have a ready recommendation for abstract algebra because I mostly didn't learn it from textbooks. I'm not all that satisfied with any particular abstract algebra textbooks I've found. An option which might be a little too hard but which is at least fairly comprehensive is Ash, which is also freely legally available online.

For the sake of exposure to a wide variety of topics and culture I also strongly, strongly recommend that you read the Princeton Companion. This is an amazing book; the only bad thing I have to say about it is that it didn't exist when I was a high school senior. I have other reading recommendations along these lines (less for being hardcore, more for pleasure and being exposed to interesting things) at my blog.