A reference to techniques for proving that the size of an integrality gap is bounded by some expression for a particular LP(or SDP, but less important) is needed. Also it would be nice to have a reference to a place where techniques for minimizing integrality gaps are described. I am new in the area of integrality gaps, which looks rather huge, so description of classical results is more preferable than that of something hot.
For sake of discussion consider minimization problem with objective function $f(x)$. Off the top of my head I can't think of any one dominant technique for proving integrality gaps. Usually the outline of the proof is the form implied by the definition of integrality gap and the details are problem specific.
For showing that the integrality gap is small (i.e. LP is good) the following proof outline is usual. Use some sort of rounding (often randomized) to construct an integral solution $x'$ with $f(x') \le c\cdot f(x)$ for every LP-feasible $x$ (and for every problem instance). It follows that the integrality gap is at most $c$.
For showing that the integrality gap is large the following outline is usual. Exhibit a problem instance with a cheap LP feasible solution and prove that there is no good integral solution.
This is somewhat heavy machinery for what you want, but there's been a large body of work on techniques for designing ever more refined LPs (SDPs) that get closer and closer to the desired integer program. A good reference that reviews these approaches is by Monique Laurent: A comparison of the Sherali-Adams, Lovasz-Schrijver and Laserre Relaxations for 0-1 programming.
Apart from that, I am not aware of a single good source of references: I assume you've at least perused the relevant chapters in Vijay Vazirani's book ?