The main textbook will be
Markov Chains and Mixing Times by Levin, Peres and Wilmer.
You can get it from AMS, or a free PDF version from the authors. There is also an errata document at the same website.
Another text we will use in the course is
Mathematical Aspects of Mixing Times in Markov Chains by
Montenegro and Tetali.
You can get it from NOW publishers, or a free PDF version from the authors.