How do I decide what problems and how many problems to do when I try to self study? I am a math major at a relatively small college with barely any choice of classes to choose from so I have to supplement my studying with a lot of self studying.
I usually have no problem getting through the chapters and understanding "most" of what is going on in the chapters. Take for example, Abstract Algebra by Dummit and Foote. I tried studying from this text during my summer break. I spent the whole summer studying the text by reading through every section from the preliminary section up to approximately section 2.3. That is I spent 4 months and barely got through 2 chapters. In my own defense of how long it took me to get through the 2 chapters or so that I got through, I solved every exercise in the text (including the exercises in the preliminary chapters) up to all the exercises in 2.3. 
My question is, how do I know when a problem is worth doing? Often, I would get stuck on a problem for hours and when I figure it out, it turns out to be surprisingly simple. Is there a time limit I should set for myself before moving onto the next exercise? If I do skip an exercise, when should I go back to it? 
 A: I think  that if you get stuck on a problem your best choice is to move on and come back to the problem later: it will probably look quite easy then.
Anyway if you don't move on you will never get anywhere because there will always be some "easy" (but who is to say?) problem that you can't do.
Mathematics is supposed to be linearly ordered but that is just a polite fiction which the Bourbaki team and its dozens of extraordinary members could not materialize in 50 years!
Nobody (and that includes Deligne and Grothendieck , two of the greatest mathematicians of all times) only uses mathematics that he has checked completely.
Dummit-Foote is a great book but you certainly don't need to know all the material in it, and even less solve all the exercises.
The contents of the book are excellent but were in part chosen by  an arbitrary decision of the authors relying on their tastes and aesthetics: you can get a Fields medal without knowing the classification of groups of order $p^3$ !
Your best option is to read what appeals to you, do a few  accessible exercises (you will have to learn to decide which ones are) and try to get to some substantial section, keeping in mind that you can come back later to any point  you want (your book will not self-destroy!)
This is professional advice: I have seen too many students in real life and on sites like this one remain stuck on page 15, chapter 1 of their book and never get to the real meat of the subject.    
Good luck and try to get to, say,  Commutative Rings and Algebraic Geometry (Chapter 15): the garden of Eden will display its marvels  to you!
[Of course you can set yourself another goal,  Galois Theory being a wonderful example]
A: If you get stuck on a simple problem for hours, then, in my opinion, the worst thing you could do is to move on to the next chapter! In mathematics, it is very hard to understand advanced concepts without a very very firm grip on the basics, so moving to a more advanced chapter without a good understanding of the basics (note: a good undestanding of the basics includes the ability to identify a simple problem when you see one) sounds like a terrible idea.
However, I do understand your question. It is usually very hard to judge your own knowledge, and whatever you will do, that will remain a problem. The best you can do is find a study partner and study together with him, having him correct your solutions and you correcting his or something similar...
A: Everyone keeps saying that math builds advanced concepts on basic concepts, but the truth is that sometimes there is more than one way to build up a concept and authors don't always clue you in as to how they're building up the concepts, so it's very hard for you to know whether in a particular exercise they're trying to feed you a concept you need to understand the next section or if they're merely pointing out some interesting sidetrack (though some authors do mark some exercises as optional).
Maybe a time limit would help you, but my advice to you is that as soon as you feel stuck on a problem, get a second perspective, such as from another book on the same topic, or from your professors. Maybe the first book fleetingly passed over some small but important detail that caused you to become confused, and maybe another book properly explains.
