Moritz Groth put up some excellent lecture notes: https://arxiv.org/abs/1007.2925v2
If you are more categorically minded, Emily Riehl's book has a lot about quasi-categories: http://www.math.jhu.edu/~eriehl/cathtpy.pdf and her website has a lot of exposition about cosmoi, an approach to $\infty$-categories via 2-category theory, and not requiring one to work in the setting of quasi-categories: http://www.math.jhu.edu/~eriehl/
