Sorry to disappoint you but there is no reference with many figures and examples, and for a good reason: Surgery theory works best in higher dimensions so most pictures are schematic at best, while examples are notoriously hard to work out, and there very few of them in textbooks (and not that many in the literature). Also surgery is at its best analysing implicit and partial information, so it is unreasonable to hope for lots of pictures and examples.
I suggest the following road map:
Homotopy theory (as e.g. in May's textbook), characteristic classes (Milnor-Stasheff), h-cobordism theorem (Milnor and Kosinski's "Differential manifolds").
Papers by Milnor and Kervaire-Milnor (see Danny Ruberman's answer).
Browder's book "Surgery on simply-connected manifolds" and Ranicki's "Algebraic and Geometric Surgery".
Wall's "Surgery theory" and Lueck's "A Basic Introduction to Surgery Theory".
Orginal sources: a huge list is being maintained by Andrew Ranicki.
Disclaimer: I am no expert in surgery theory, but I have been studying it for many years and finally got to the point of using it for Riemannian geometry purposes.