Random matrix theory is a diverse area; and different people prefer different introductions. One question is whether you're mostly interested in mathematical aspect or in applications in physics (and other areas). But I can recommend everything below.
Topics in random matrix theory,
excellent lecture notes by Fields medallist Terence Tao
Introduction to the Random Matrix Theory: Gaussian Unitary Ensemble and Beyond,
lecture notes by Yan Fyodorov
Topics in random matrix theory
lectures mostly focused on QCD by Jac Verbaarschot
Random Matrices, by Madan Lal Mehta
canonical on the orthogonal polynomial approach
An Introduction to Random Matrices, by Greg Anderson, Alice Guionnet and Ofer Zeitouni
Overview of applications
The Oxford Handbook of Random Matrix Theory,
Editors: Gernot Akemann, Jinho Baik, Philippe Di Francesco.