First of all, make sure to have a few references on category theory available. Good ones include:
It also pays of to learn about the insights leading to $\infty$-categories before learning about their theory proper. A good reference here is John Baez's An Introduction to $n$-Categories. Another one is Section 1.2 of Finnsen's thesis.
$\infty$-Categories require two fundamental prerequisites: model category theory and simplicial sets.
Simplicial Sets. Friedman's An elementary illustrated introduction to simplicial sets is a marvelous introduction for beginners. For more in-depth references, there are May's Simplicial Objects in Algebraic Topology, and Simplicial Homotopy Theory by Goerss–Jardine.
Model Categories. Good references for model category theory include:
4. ($\infty$-Categories, finally)
It is hard to capture in a precise way the idea of an $\infty$-category as a set of objects, together with a set of morphisms, a set of $2$-morphisms, and so on. There are two ways of approaching this difficulty, one traditional, the other very recent.
Via Quasicategories. The traditional one is to use models for $\infty$-categories. One such model is given by a special kind of simplicial set called a quasicategory. This is the approach developed by Joyal and Lurie. For learning the theory of quasicategories, there are:
Via $\infty$-Cosmoi. The second one is the model-independent approach of Riehl and Verity (which is currently being developed). Instead of axiomatizing what $\infty$-categories are via models, Riehl–Verity axiomatize the mathematical object in which $\infty$-categories live, and call it an $\infty$-cosmos.
When working with an specific model for $\infty$-categories, one is often lead to complicated arguments involving its combinatorics. On the other hand, in Riehl–Verity's framework, it is possible to prove statements about $\infty$-categories in a much simpler, model-independent, way
Riehl and Verity are currently compiling their work in a textbook, called Elements of $\infty$-Category Theory.
Background on model categories and simplicial sets