Of the four things you mentioned, the prerequisites for analytic number theory can be brutal. Setting up the $\zeta$-function, for example, isn't particularly difficult, but you'd probably need a background in complex analysis to cover standard results like the prime number theorem and Dirichlet's theorem. (The basic idea is that the poles of the $\zeta$ function control the growth of primes, but you need some sort of Tauberian theorem and some familiarity with complex analysis to extract an actual proof from that idea.) Number theory is a lot of fun, but there's a lot of machinery (in algebraic or analytic number theory) required to get to interesting results.
General probability can involve pretty deep measure theory, but the probabilistic method may not be so bad if you're in the finite category. Most of my experience with it is in contexts like hyperbolic groups, so I can't say much about it from the perspective of someone coming in from a contest-math-style background. If it interests you, though, go ahead and give it a try. The worst-case scenario is that you decide it isn't for you and move on to something else.
Group theory is pretty broad. There are standard texts like Artin and Hungerford, though you might be more interested in combinatorial group theory. If you're familiar with the basics already, you might want to look at representation theory. It's a great subject, and there are lots of sources (I'd recommend Fulton and Harris) that are well-written and accessible.
Topology's another great choice, though the problem there is that point-set topology is tedious but necessary before getting to the good stuff. Munkres is the standard text for it, but it's a pain to slog through. Books like Bott and Tu or Hirsch are great for the differential case, and Hatcher is beautifully written (and available for free online) for the algebraic case, but both have prerequisites that you might not be familiar with already.
