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.
 A: 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
A: Ring theory works with commutative groups, certainly in the early stages - fields are commutative, polynomial rings are commutative - you can do a fair amount without having to deal with the non-commutative case (think of the large text books on commutative algebra). This is a major simplification. 
Also, as others have mentioned, there are some very familiar canonical examples of rings.
A: 
Is it possible to learn ring theory if one's familiar, but not good at group theory?

Absolutely yes; your study of Dummit and Foote should give you enough of a foundation in "group theory" to successfully study "ring theory" in greater depth. So it is indeed possible to learn ring theory before being fully proficient in group theory.
But I also want to add: 
It is far too early for you to conclude you are not good at group theory! You're an undergrad, as you point out, and your experience when first encountering any new theory or topic is not really sufficient evidence to determine how "good" you are at it, especially if you're making that judgment based on your progress through one book's coverage of groups.  
Every serious student of math encounters a wall at one point or another (and for most of us, many many times). Sometimes the things that first "trip us up" are precisely the things that end up fascinating us. 
Furthermore, try not to judge your mastery of any topic (e.g. group theory) based on its presentation in only one text. (In my humble opinion) Dummit and Foote's text is not all that great in covering groups. 
So explore a bit by supplementing Dummit and Foote's coverage of groups with other resources: e.g., find a text or two that approach groups differently:  


*

*Artin's Algebra covers groups well; see especially chapter 2. 

*Or - if you're really struggling with groups - try looking at Fraleigh's A First Course in Algebra, which does great with introducing groups and motivating the material. The book is written in a way that is very readable, intuitive, and includes a lot of examples.

*If you can't access one of the above texts through a library, and are looking to limit expense, J.S. Milne has a nice site for course-notes, including a ~$140$-page pdf on Group Theory.
So try not to "write off" group theory quite yet; you may find you like group theory, after all!
A: It is certainly possible to study rings before groups. In fact, this is the approach taken in say Shifrin's Abstract Algebra as rings are perhaps more natural than groups to some.
On the other hand, the study of rings will involve some group-theoretic results, but these can always be picked up on the go.
A: I think this is the topic in which courses differ the most... Some courses start with rings because it's the historical way, starting with Galois works on rings based on his tudies about polinomials. Groups appeared after that. Some other courses start with groups going not the historical way, but the formal way, so when you get to rings you can use the knowledge you have about groups. So, as they have said, absolutely you can study ring theory with little knowledge about groups, as many textbooks dont presume any knowledge.
A: I must say I don't see why you are interested in ring theory if you are not good at group theory ; you don't give interest in a subject just because it looks cool, but because you want to be able to use it in some way. If you just want to read about it, you can, whether you are good or not, it's not a problem. Your eyes will still open and you will still be able to read the book! The issue is when you will want to do exercises ; group theory and ring theory are theories among the algebraic theories, so their nature is very similar ; it is very likely that being good/bad at understanding one will imply being good/bad at understanding the other, although it is possible that you feel a little bit more comfortable working with objects with a little bit more structure than groups. 
As amWhy points out, we interpret your "not being good at group theory" as "you're not comfortable with it". You might turn out to feel comfortable with it later and be very good at it.
If this is because you want to read about a ring theory course you must take before you finished your group theory class or something like that, I think it is a bad idea ; the Dummit & Foote book is a good book to use as a reference and is full of relevant exercises, I am a fan of it myself, but I must say I did not begin learning group theory using this one, it lacks a little bit intuition. For books that build intuition on groups in a more relevant way, I could provide you a reference. 
I'll add that when one begins ring theory, the first examples of rings differ a lot from the group-theoretic ones, mostly by the fact that they are infinite. A lot of the elementary group theory focuses on finite groups because they are easier to study ; in ring theory, the basic examples would come from subrings of fields ($\Bbb Z$ or $n\Bbb Z$ would be an example), number rings or function rings, and their quotients. This gives them a quite different feeling from the elementary examples in group theory such as $\Bbb Z/n \Bbb Z$, $S_n$, $A_n$, $D_n$, etc.
To sum things up : give it a try and see what happens, if you have the time to do so.
Hope that helps,
