Michaela has just acquired one of these puppies, and her old man, who missed the craze in the 70’s, is pretty much just as lost as she is. Any sites where they teach you?
There are several different solutions. The layer-by-layer method is pretty simple but slow. The corner method is a bit more difficult but a lot faster.
If you turn it the right way, the cubes can be removed from the assembly and reassembled. Quick and easy. 
I bought a book. Then again, it was 1982. 
Just build a robot to do it for you.
I solved it when it came out the old fashioned way- hours of trying.
I would find a way to get one aspect, row or column, changed from one thing to another, and made up my own shorthand for how 1U for first row up, 2R for middle row right, etc.
Then keep at it until you get closer and closer and can see the end.
I burned a lot of free time to do it, but I eventually figured out an algorithm to solve the cube. I should say that I made frequent use of the disassemble-and-reassemble trick mentioned by RealityChuck. That way, I could try an experimental sequence of moves on a pristine cube to see their exact effect without taxing my memory.
My algorithm is not anywhere near the optimum. I could never get it under a couple of minutes. Competitive solvers are just ridiculous. Here’s someone solving the cube one-handed in under 18 seconds, evidently setting a world record.
The colored patches could also be peeled off and restuck.
That’s nothing. If you’re going to build a robot, make it out of Lego.
The key to solving the Cube is to learn sequences of moves that move some parts around, but leave others untouched. Then, figure out what order you’ll do the pieces in. With the method I use, for instance, I solve all of the edge pieces first (making plus signs on all of the sides), then I do the corners. So first, I need a technique to get the first several edges (this is easy, since it doesn’t matter what else I mess up at this point). Then, I figured out some tricks to get the last few edges, without messing up the ones I already had. But I could still mess up the corners as much as I want, because I hadn’t solved them yet anyway. Finally, I found (actually, learned from a classmate) a sequence of moves which would shuffle a few corners around, without disturbing any edges, and repeated applications of that sequence (or variations on it) would finally get everything solved.
I learned it from a book, the layering method. First the four side pieces of one face, then the four corners for that face. Then the next layer - the four side pieces adjacent to those corners. Then the last four side pieces (on the “top” face) and then finally the last four corners.
I could do it in about 2.5 minutes when I was in the 8th grade. I don’t think I could do it anymore.
By swapping two stickers on a pair of corner or edge pieces, you can make the cube unsolveable. This is a great prank to play on a competitive cuber.
Spin the middle side topwise. TOPWISE!