I broke my last Rubik's cube some months back. I'd just completed learning to solve it by that time, when the cube came apart in my hands. I tried putting it back together, but one of the center pieces was broken. The assembly of the cube is itself quite interesting. All the parts sit perfectly on top and beside each other, with a minimal scaffolding at the center ; and 'course there's more than one way to assemble it, meaning you can basically create a unsolvable cube (Now, there's a trick to play on someone). Anyway, I got my new cube a few days back, thanks to a friend from the US. It looks very rigid, just like the last one, and initially takes some time to loosen it ; but after that it's smooth sailing all the way, that is of course, if you know how to solve it. There are loads of tutorials on the web to get anyone started on this. But initially, the best way would be to just mess around with it a bit, try getting a particular piece to a particular place and such; get used to moving stuff in three dimensions as opposed to 2-D jigsaw puzzles.
The most basic approach to solve it uses layering. The 3x3 cube is split upto into 3 horizontal layers, and one goes about solving a layer at a time. The first two layers are pretty easy and basic intuitiveness is sufficient to complete them. The last layer is the most difficult of them all, partly because one has very little room to work with without disturbing the completed layers. If you're really good and patient at it, you might figure it out by yourself, or if you're like me, then search for the easiest algorithm, and solve it that-a-way. Either way, after some time, the modus operandi will gradually evolve right into the cuber's hands and a feeling of accomplishment is inevitable. This is a really good tutorial to get started with, and frankly I don't want to get past it. I'm just a casual end-user of the cube, not a geek trying to speed-cube my way into 16 seconds. Of course, if one is really good at it, then the 4x4 Rubik's Revenge is definitely the worthy adversary.