I'm certainly not an expert, but I'll try to give some advice.
For game theory proper there's Yvain's sequence (and Schelling's book, which it's based off of) and/or Tadelis's Game Theory.
A good way to get to mechanism design might be through introductory economics and auction theory. McAfee's Introduction to Economic Analysis is an open econ textbook, good for people with a solid understanding of basic calculus. It assumes this bit of math so that the presentation is a lot shorter and more elegant. (Apostol is my calculus textbook of choice. If you've never done math where you actually have to prove things, then Velleman's How to Prove It will get you started. If you can't prove then you're just memorizing passwords. It's easier than it seems at first.) After IEA, Krishna's Auction Theory will segue from basic auction theory to basic mechanism design. Haven't gotten much further than that.
There's also a mechanism design sequence on LW. I haven't looked at it too closely and it might move too quickly for someone without the right background.
This is an awesome answer, thank you very much!
IEA is also a favourite of mine.
There are several well-known games in which the pareto optima and Nash equilibria are disjoint sets.
The most famous is probably the prisoner's dilemma. Races to the bottom or tragedies of the commons typically have this feature as well.
I proposed calling these inefficient games. More generally, games where the sets of pareto optima and Nash equilibria are distinct (but not disjoint), such as a stag hunt could be called potentially inefficient games.
It seems worthwhile to study (potentially) inefficient games as a class and see what can be discovered about them, but I don't know of any such work (pointers welcome!)