So the WS2812 LED controller is a very standard part, and it is well supported in the Arduino ecosystem. Tweaking4All has a very pretty page all about it, and the interfacing couldn't be simpler.
One 470 Ohm resistor and a 5 volt power supply, and you've got an LED strip!
But how to control it? Well, it uses a bit of a timing dependent method of clocking out the bits that I haven't bothered to learn, so I just use the lovely NeoPixel library from Adafruit. They even have a great PDF manual that gives you all of the technical details you need. Again, there are plenty of examples on how to use it, so I greatly suggest trying a few of those.
I guess I've done about as much research as I need to do at this point, so the next step is for me to actually order some parts and stick them all together so that I can start writing some code!