FLCL was the first anime I watched; I'd never really had an interest in anime, but I did like The Pillows. And when I found out that the soundtrack to this was made mainly of their songs, it was a must watch for me.
I've seen reviews that say that FLCL is so crazy and out there that it's a bad place to start with anime as it can be so confusing that it might put you off, but I didn't find it that weird to be honest. I admit I'd be lying if I said it made perfect sense instantly upon first watching, but it certainly wasn't offputting. If anything I found it intriguing; it's so fast paced and insane that it kept me interested the whole 3 hours the series lasted for. At the most basic level it could be described as a coming of age story, but that description doesn't really do it justice. The plot is really open for interpretation; to younger viewers it's just an awesome story about a bored kid who's life is transformed by the sudden appearance of aliens, robots and all sorts of other crazy happenings, but viewed from an older perspective the plot is full of innuendo and symbolism that adds a whole new layer to the show.
The style of animation changes constantly, always giving you something new to see or at least a new way of seeing it, the dialogue and characters are hilarious throughout and of course, the music is excellent (obviously, I mean it is The Pillows).
Overall this makes for one awesome anime; the animation is as superb as always with Gainax productions, the action scenes are intense, the funny parts had me laughing out loud frequently, the plot has an insane ammount of depth and metaphor to it and the voice acting is excellent in both the dub and sub versions (although having originally seen it with subtitles I still prefer it that way, the English voice for Haruko just doesn't seem right to me).
So basically if you're the kind of person who's a fan of stuff that looks good, sounds good, is funny or any combination of those three I'd highly recommend you watch FLCL. :)