It really depends on whether you want something a little more encyclopedic, or something with a different goal.
Here are some suggestions:
Encyclopedic. These are books that have a lot of stuff on a lot of topics, but generally do not delve very deep on any particular subject. They will have pointers to further work, though:
Joseph Rotman's An Introduction to the Theory of Groups; 4th Edition is from Springer. It starts pretty much from zero, but you should learn some new stuff from it about $G$-sets, and maybe even the Sylow Theorems. Once you hit Chapter 5 (of 12), it will probably be mostly new to you.
Lang's Algebra, revised third edition from Springer. The encyclopedia of Algebra. In my opinion, not a great book to learn from, but a great reference. It is not exclusively Groups, though, and quickly veers into Fields and other stuff.
Hungerford's Algebra is similar to Lang's.
Derek Robinson's A Course in the Theory of Groups, second edition, from Springer. Starts with the basics, but it hits Free Groups and presentations in Chapter 2 (of 15) and continues from there. It is a bit more focused on infinite groups rather than finite groups.
Kurosh's The Theory of Groups. There are two volumes, link is to volume 2, and I spotted a "free PDF" on a google search.
Somewhat more advanced, with a particular point of view
Peter Neumann, Gabrielle Stoy, and Edward Thompson's Groups and Geometry from Oxford Science Publication, is more focused on groups as measurements of symmetry, with a good chunk on actions and the connection between groups and geometry, as the title might imply.
I. Martin Isaacs, Character Theory for finite groups is the book to learn group characters from (it is related to representation theory). From Dover, it is fairly inexpensive.
I. Martin Isaacs, Finite Group Theory, is a kind of "A second course in group theory". It starts assuming you know basic group theory and goes into the Sylow Theorems (with a couple of interesting extensions you don't usually see). The goal of the book is to give a representation-free proof of Burnside's $p^{\alpha}q^{\beta}$ theorem. It is published by the AMS in their Graduate Studies in Mathematics series.
Magnus, Karrass, and Solitar's Combinatorial Group Theory is the classic book on combinatorial group theory.
Specific to group theory in topology
You mention applications of group theory to topology. There are a couple of books that deal with that directly.
From a more topological point of view, we have John Stillwell's Classical topology and combinatorial group theory.
For one of the classics, Max Dehn's Papers on Group Theory and Topology. Keep in mind that these are the papers he published. Influential, but possibly hard going to begin with.
There are a few other books that might contain interesting stuff depending on your interest. I'm afraid I don't have one that deals specifically with group actions. Some of the older books might contain interesting stuff that is not done much any more (Marshal Hall's classic book "The Theory of Groups" has a bunch of stuff on the Burnside problem and Jordan's theorem on multiply transitive groups). Some have specific topics in mind, such as Susan McKay's book on $p$-groups, or Hanna Neumann's classic book Varieties of groups, J-P Serre's Trees for groups acting on trees, etc., but those are the kind of books you should search for later, after you've decided you want to pursue this. If you want to work in abstract algebra, you'll probably want to get Lang or Hungerford at some point.
My recommendation if you are not wedded to looking at group actions is Isaacs' Finite Group Theory. If you definitely want to look at group actions, look at either Neumann, Stoy, and Thompson's book, or the relevant chapters in Rotman. But that's just my opinion, and I'm not a very big "geometry" guy, so someone else might be able to better advice you there.
