Dear James,
To a large extent, the answer to this question will depend on how successful you are at working on your own without the infrastructure of a course/lecturer/problem sets/etc. to guide you. Given this, if you don't know yet whether you work well by yourself, there's only one way to find out: try it! You may find that you are good at working by yourself, and, if so, it doesn't really matter what your background is: you can fill it in by reading more books. On the other hand, you may find that it's hard to make progress without the usual structures that a course provides, and that's fine; many successful mathematicians were not all that independent when they were undergraduates.
One book that you can read which doesn't require much background at all is Hardy and Wright's classic text on number theory. It does not suit everyone's taste, but if you are not yet sure where your taste lies, you can take a look and see if you like it.
One thing that you didn't address in your post is the question of how comfortable you are with reading and writing proofs. If you are not comfortable with this aspect of mathematics, then my suggestion of Hardy and Wright won't be terribly appropriate, and neither will many of the others. If you are comfortable with proofs, then in some sense there is no limit on what you can do by yourself, since (at least in principle) you can pick up any textbook and try to learn what is in it. On the other hand, if you find that you aren't (yet) comfortable with reading and understanding formal proofs by yourself, then it will be harder to go very far by yourself, and it might be better to focus on your formal course work for now. (And, if your ambition is to pursue pure mathematical research, you should try to take courses that introduce you to reading and writing proofs as soon as you can.)
Whatever your situation is, you should always be sure not to neglect your formal coursework (even if the work you are doing on your own turns out to be more exciting). Excellence in formal coursework is more or less a requirement for going on in graduate school, which is in turn a requirement for becoming a research mathematician.
Regars
Regards,
Matthew
