You know the routine. You come across a topological space X and you need to find its fundamental group. Unfortunately, X is an unfamiliar space and it's too difficult to look at explicit loops and relations. So what do you do? You look for another space Y that is homotopy equivalent to X and whose fundamental group is much easier to compute. And voila! Since X and Y are homotopy equivalent, you know that the fundamental group of X is isomorphic to the fundamental group of Y. Mission accomplished. Below is a list of some homotopy equivalences which I think are pretty clever and useful to keep in your back pocket for, say, a qualifying exam or some other pressing topological occasion.
Y. Smirnov. General Topology and its Relations to Modern Analysis and Algebra: Proceedings of the second Prague topological symposium, 1966, page 332--340. Praha, Academia Publishing House of the Czechoslovak Academy of Sciences, (1967)