My favourite introduction to stochastic processes and stochastic calculus is the book Stochastic Calculus by Paolo Baldi. It is very clear yet precise at the same time, and comes with hundreds of exercises with full solutions prepared. It hits all the important topics - conditional probability, Markov processes, martingale theory, stochastic calculus with respect to Brownian motion, and SDEs. If you want to go further after that, the book Brownian Motion and Stochastic Calculus by Karatzas and Shreve is a nice second read. These two books should already take you quite far as far as general knowledge goes, but do let me know if you want references for more specific topics! Edit: I should say that stochastic analysis is not the only type of “advanced” probability. There are far more topics, however it cannot hurt to have a basic background in stochastic analysis.