There's no substitute for a good mathematical intuition, especially as you tackle the more difficult problems. But having a standard bag of tricks up your sleeve can be very helpful. Just use them as a tool, not as a crutch.
If you're interesting in this sort of thing, I highly recommend working your way through the Art of Problem Solving texts (www.artofproblemsolving.com).
Most of the higher level math text books I've read present theorems and their proofs , but not much on strategies for actually solving problems (a notable exception: Wilf's Generatingfunctionology), so this seems like it should become a very nice resource.
As one would expect from a brand new site, it's rather short on content. The content that's there right now, though is pretty good.
If you're interesting in this sort of thing, I highly recommend working your way through the Art of Problem Solving texts (www.artofproblemsolving.com).