Bass-Serre theory textbook

As mentioned by Andy Putman in the comments, the classical (and probably the best) references are Serre's book Trees and Scott and Wall's paper Topological methods in group theory.

Serre's approach is elementary and essentially self-contained, based on combinatorial arguments. Scott and Wall's approach is based on algebraic topology (in particular, they use covering theory). If you are already familiar with some algebraic topology, then you may find Scott and Wall's approach more intuitive, more visual. Personally, I began by reading Serre's book, but I prefer the point of view of graphs of spaces.

You can also take a look at Henry Wilton's blog Geometric group theory. There are several blogposts dedicated to the Bass-Serre theory. In complement to Serre's book, an interesting complement is Higgins' article The fundamental groupoid of a graph of groups. Looking at the fundamental groupoid instead of the fundamental group makes some arguments more natural.