As a graduate student I was taught homotopy first (including higher homotopy groups), then singular homology, and then cohomology. The instructor was quite good, but now I feel that the order of presentation was backwards.
I think starting with homotopy is fine as long as you stay in low dimensions, but degenerates into algebraic nonsense otherwise. I highly recommend Stillwell's book Classical Topology and Combinatorial Group Theory where he takes this approach.
Edit: I am not a topologist. I am probably further from being a topologist than people who have left similar disclaimers.