I would have to disagree with pretty much all recommendations here. I've come to believe that becoming a "good" music fan and listener you actually have to practice. You have to listen to a lot of music and you have to find out what you like and don't like.
Eventually you will either kind of taper off and come to realize that maybe you're just not as interested in music intellectually as you thought you were, or you'll start to really develop a taste. You'll develop your own pallet for music. (And something tells me I'm using the wrong spelling of pallet here, but fuck it.) And I think anyone with a real taste for music will usually naturally move into some form of indie music anyway, so no worries there.
And honestly just taking all of the recommendations from this thread will probably just result in you trying to like the bands for a while before you eventually realize you don't really like them that much.
That being said, I personally think Queens of the Stone Age is better than any of those mainstream, faux-indie acts out there that could lead you into indie, and personally, the Queens are where my own musical taste really started to develop (and also one of the few bands I listened to when I was like 15 that I still think are the fucking bee's knees). So that's a group I have to recommend to any young person unacquainted with more obscure music. The Queens and The Beatles. But again, that's just me.
Edit: To add some more explanation. I think there are a lot of popular acts that are genuinely good and unique enough that they will help you develop more refined musical taste. Essentially giving you some "practice" in listening to music. Or for a different kind of metaphor, they're like a gateway drug to more extreme or difficult shit.