Is it possible to learn ring theory if one's familiar, but not good at group theory? Is it possible to learn ring theory if one is familiar with but not good at group theory?
Background: I’m using Dummit and Foote's Abstract Algebra, and I am an undergrad.

Personally, I think "Dummit & Foote" is excellent for ring theory. And much more accessible for a beginner than groups. (Dummit wrote rings, Foote groups.)
To build more confidence in groups, the chapter in Artin's "Algebra" (go for the 2nd edition) on groups is very well written and can give a very intuitive understanding of what are key points.
You can also watch the specific videos on groups in this excellent lecture series by Benedict Gross at Harvard, that follow Artin. You will most likely want to watch the ones on rings as you study that topic. 
http://www.extension.harvard.edu/open-learning-initiative/abstract-algebra
