I would agree with Henry T. Horton that, while stating that "we do assume familiarity with the elements of group theory...", the material relevant to continuing on in Munkres is listed/reviewed at the beginning of the section on fundamental groups:
- normal subgroups;
- quotient groups;
with much of this inter-related.
Fraleigh's A First Course in Abstract Algebra would be a perfect place to learn these basics of groups and group theory; the text covers most of what is listed above in the first three Sections (Numbered with Roman Numerals) - the first 120 pages or so, and some of the early material you may already be familiar with.
It's a very readable text, lots of examples and motivation are given for the topics, and with very classic sorts of exercises. This should certainly suffice for what you'd like to better your chances of conquering "Part II" of Munkres.
A good resource to have on hand while reading Munkres, and/or to begin to review before proceeding with Munkres, is Michael Artin's Algebra (2nd edition). It does a great job treating groups!
If cost is an issue, and/or you can't find a copy of Fraleigh's text at a library:
you might also want to check out Beachy and Blair's On-line Abstract Algebra Text (access to the second edition) and focus on the material covered through/including Groups.