Give yourself about 20 hours to learn and play around, by then you'll have gotten the hang of it and will probably be able to solve most arrangements with ease.
NO! Solving the cube is boring, discovering how to solve it is not. The journey is the reward. What you should do is learn a tiny bit of algebra. In particular, learn about commutators (http://en.wikipedia.org/wiki/Commutator)
Then, experiment with commutators on the cube to look for ones that have very localized effects. Those are those 'algos' others want you to learn from a book. For example, check the simplest commutator LTL'T' and figure out how to use that as part of a larger commutator that permutes the three corners of a cube.
Yes, it will probably take more than those 20 hours (that can be done faster, I think. You need at most 4 tricks (permute three corners, permute three edges, flip two corners, flip two edges should do), each maybe ten turns to solve any cube), but it will be more fun.
It took me way more than 20 hours to solve my first cube, but the first two times I did it, I didn't even know how to solve it yet.
Couldn't agree with this more. It took me a year of chipping away at it to come up with a solution. Finally finished it the first time while my flight was touching down in thailand.
I've heard mine is not the normal algorithm people are taught. The one I use is totally suboptimal - but that's a wonderful thing. There's forever more to discover. It's actually a brilliant problem for the types of people that haunt HN. I implore those of you who have never tried to give it a shot. And remember, if you look online, you're only cheating yourself.
Approach it very methodically. Learn the algos. Practise. And that's really all. Here're two videos to get started: http://www.youtube.com/watch?v=mvhKNXHQqyM - http://www.youtube.com/watch?v=MaltgJGz-dU
Give yourself about 20 hours to learn and play around, by then you'll have gotten the hang of it and will probably be able to solve most arrangements with ease.