If you wish to become a solid mathematician, there are certain topics with which you will want to have familiarity, even if you do not intend to delve deeply into them. For instance, most research mathematicians will have had a decent training in point-set topology and will have learnt at least a few of the major theorems and techniques in the area (Urysohn's lemma, Tychonoff's theorem, Urysohn's metrization theorem, partitions of unity etc.).
Ultimately, the extent to which you remember these results and techniques does not determine your quality as a research mathematician. However, open sets and their theory are ubiquituous in nearly every branch of pure mathematics, and having a feel for certain topological concepts is certainly desirable. (Perhaps not essential, however, depending on which branch of mathematics you pursue.)
Again, this is not to say that someone can be dismissed for not knowing point-set topology: there are plenty of ways one can do meaningful research without having a training in point-set topology on the magnitude of Munkres' Topology: A First Course or Kelley's General Topology, for example, and there are at least a few professional mathematicians who demonstrate this.
As for Sylow theory, the topic, of course, falls into the area of finite group theory. (I hasten to add, however, that Sylow theory does have applications to the theory of locally finite (but possibly infinite) groups.) While finite group theory is an exciting subject full of rich structure and powerful theorems, I suspect that there are not too many branches of pure mathematics where the methods of this theory are instrumental to doing meaningful research. One notable exception (to some extent) would be algebraic number theory. For instance, the principal ideal theorem of class field theory can be proven using the techniques of transfer (in group theory). Also, it would be fair to add that most algebraists have a solid training in finite group theory, even if their research interests lie in other aspects of algebra.
Succinctly, I think that it is fair to say that being comfortable with the various techniques of group theory and topology, whether or not you pursue either of these subjects, can be helpful in many areas of mathematics. The subject matter in its exact form may not repeat itself in other areas, but the techniques, ideas and intuitions may do so.
