I am currently experimenting with GP and I wanted some test problems that have already been solved with GP. This way, I would know that genetic programming would be able to provide a solution that works. Also, shorter and simpler test cases would be better since I'm planning on running the algorithm many times. Thank you.
Here's a list of the more common benchmark problems GP is tested on. These are very widely used and descriptions of each should be easy to find with Google. For a very thorough description of each, including suitable fitness functions and results you can compare against see Koza's first Genetic Programming book if you can (the field has come on a bit since then, so they're not leading results, but you're unlikely to find more thorough descriptions).
Boolean Functions
Symbolic Regression
Path Finding
Control
Iterative/Recursion
There's also a whole range of other problems that GP has been successfully tested on, here's a few more, although these are perhaps beyond what you're looking for at the moment:
If you're just looking for a few problems to get started with, to see the algorithm working and producing solutions, then I'd suggest starting with a simple symbolic regression (where you're trying to find a mathematical formula), then try one of the simpler boolean problems such as Even-3-Parity or 6-bit Multiplexer. The good thing about these problems is that you can easily ramp up the difficulty by changing the formula you're trying to find or increasing the number of bits.