Morse theory is an example of such method. The classification of compact surfaces using Morse theory is done for example in the book of Hirsch, Differential topology. The book of Milnor, Lectures on the h-cobordism theorem goes a step further by proving the Poincare conjecture in dimension bigger than five using Morse theory.
It should be emphasized however that "look at $F(M)$, the set of functions on $M$" is just one way to understand a space $M$ amongst many others.
