You don't want to learn to play an instrument, you want to be able to play an instrument. It's really hard to turn a preference for a future where you can jam with friends into actually getting yourself to sit down for daily scales practice. Most people get sick of it and give up.

The standard way to teach an instrument is based around the idea that time is the limiting factor and you should be spending it as efficiently as possible. You practice on scales, rhythm excercises, and easy pieces to get the most out of the minimal time you can put into practicing. But this isn't how it works for real people: if practicing is somewhere between boring and unpleasant, you give up or grudgingly force yourself to do a small amount. Replace 'practicing' with 'playing around with the instrument' and get to the point where you're enjoying yourself, however, and time available increases dramatically. You'll spend your spare time on the piano instead of the internet.

The game is interest management: keep playing interesting for yourself and you'll do lots of it. Music is a field where time spent matters much more than how it's spent.

Some people will tell you that it's important to have a teacher so you don't learn bad habits that will be hard to correct later. I say don't worry about it, have fun, and do what works. You can adjust your technique when you start finding it limiting, but if focusing on technique makes you lose interest you'll never get far enough to benefit from having good technique from the start.

Update 2012-09-11: Julia's learning the banjo, and this approach seems to be working for her.

[1] Anyone can sit down at a piano and start fiddling with it, letting their learning system put the pieces together. Sit down with a flute and you'll be exhausted before you manage a breathy whistling tone. Learning curves matter. Start with an instrument where you can get a decent sound for a single note (any note) quickly. What are these? In rough order: piano, drums, accordion, guitar/mandolin, bass, pennywhistle/recorder, trumpet/trombone/tuba, violin/viola, flute, sax/clarinet, oboe. If you do want to learn one of the trickier ones, you might do better to get someone show you how to make enough of a sound that you can let experimentation and play take over.

[2] Also fiddle, octave fiddle, recorder, and penny whistle. I'm currently working on trumpet. I did learn guitar as a kid with standard lessons.