I would like to suggest a book that we used for a first semester course in probability.
The book "Introduction to Probability Theory" written by Paul G. Hoel, Sidney C. Port and Charles J. Stone is a well-written book on probability theory. There are a lot of exercises that gear you through. If an exercise is a bit hard, they also leave a hint for the reader. (Amazon Link)
Prerequisites: (Suggested by the authors in the preface.)
- Elementary Calculus that includes multiple integration.
I would also suggest that one has an elementary knowledge of counting. This would help cover a few sections in Chapter-1 rapidly. It emphasizes the more important concepts, while striking a delicate balance between mentioning those results that require sophisticated machinery to prove and those easy to prove while not very important.
One good thing about the book, it mentions some results whose proof requires measure theory, without omitting them. Exercises are carefully chose and well-organised through the text.
Hope you enjoy reading it. (And, you may find some good places to look without having to buy. I'll stop here in this regard.)