Usually textbooks list problems/exercises in the order in which they progress from easier to solve to greater and greater levels of difficulty. So your best bet, if you don't want to solve them all!, is to pick a few earlier problems, to get warmed up, then some middles problems, and build up to selecting some of the problems from the last third.
A good approach would be to do all even problems, or all odd problems, or, if you aren't struggling to much with a particular section of the text, perhaps every third problem. That way you are following a progression from starting with easier exercises, and progressing to more difficult ones.
I'd suggest scattering a few of the earlier problems (every third problem, say), but putting most of your energy into the last half of the exercises: Those are the exercises that will ensure (or require) that you "really know your stuff"!
At any rate, save a few problems for giving your self "examinations" periodically, for the sake of personal accountability: In a typical semester-length (4-5 months or so), there is at least one midterm, and a final exam. So you might aim, say, for testing to return every month or two, first spending a little time reviewing the material covered during that time (studying for your exam), and then returning to some of the problems you left "unsolved" to "test yourself". If you find yourself struggling to complete those exercises, you might want to spend a bit of time reviewing the relevant material in order to recall (and/or deepen your understanding of) what you need to know to solve them, before moving on.
Then, "back to the books" and moving forward from where you left off!
ADDED: Here is a syllabus from an abstract algebra class taught last fall at Standford University. It includes assigned exercises from the class text (Dummit and Foote), and most of the assignments have solutions available for downloading.
You can also try googling "Dummit and Foote: .edu" to search for other links to course syllabi in which Dummit and Foote is the class text.