The probability class I recently finished (taught at an upper-undergraduate or lower-graduate level) used the text by Grimmett and Stirzaker. I really disliked this book.
I am familiar with measure theory, so it is fine if the book is measure-theoretic. However, I want to make sure the book doesn't neglect to provide a clear explanation of a the basic concepts of probability, and provide exercises for the basic problem-solving techniques. Also, I always love texts that have good motivation and intuitive explanations for things (I guess I prefer some motivating discussion rather than a totally formal text like Rudin, or arguably Ahlfors).
I think model texts for what I'm looking for are Spivak's Calculus and Dummit and Foote's Algebra. What I love about these texts are the large amount of (enlightening) exercises.
Update: Still looking for a good probability text. Feller's text has come up as a suggestion. Does that one have good exercises?