I learned probability from Grimmett & Stirzaker, Probability and Random Processes. It has a lot of exercises with a good mix of difficulties. It was a standard fixture on the desks of quants at the bank where I used to work. It's pleasant to read, includes interesting applications, does its best to build intuition and the occasional joke is pretty funny (YMMV).
Caveats: I stopped just short of the material on the Itô calculus, and if/when I come to study that subject I'll probably seek out a more leisurely treatment. Also, although it does things in terms of sigma-algebras etc the book aims to teach probability, not rigorous measure theory.