I like the book Brownian Motion -
An Introduction to Stochastic Processes
by René Schilling and Lothar Partzsch:
* As the title of the book suggests, it concentrates on Brownian motion which is, without any doubt, the most famous and most important stochastic process (with continuous sample paths). It discusses path properties of Brownian motion, presents several ways how to construct Brownian motion and introduces stochastic integrals with respect to Brownian motion. Moreover, it contains two chapters on stochastic differential equations as well as a chapter on the connection to PDEs.
* The book starts right from basic definitions and properties; the reader should be familiar with measure/probability theory and the basics of (discrete) martingale theory.
* There are full solutions to all exercises available on the web.
