@JKgo, there are a lot of anime out there. Most of them are terrible, especially in the case of anime produced after 2007 for some reason. They put that "fanservice" rubbish in to try to keep viewers (and therefore keep making a profit) even though the shows they are making are terrible. But then again, all of that gave us Koyomi Arararararagi's wonderful, wonderful abs.
Oh wait. Yeah.
Anime recommendations. Hmm...
If you can put up with a small amount of rather poorly judged "fanservice" and a few molestation jokes that quite frankly were in rather poor taste, The Melancholy of Haruhi Suzumiya is a great anime, with a witty, comedic script, great characters, an entertaining urban fantasy setting and great animation. Just make sure you watch it in the chronological order.
If you haven't seen Neon Genesis Evangelion, then you should watch Neon Genesis Evangelion. You might love it, you might hate it, but... It's Eva. You kind of have to watch it if you like anime.
I'm going to lump Death Note and Steins;Gate in together as one recommendation because honestly they offer basically the same thing. A well paced sci-fi or fantasy psychological drama thingy with a hammy protagonist. The difference being that Death Note has a better setting and stuff like that while Steins;Gate is tongue in cheek rather than unintentionally funny and doesn't have a terrible second arc. Or an inconsistently characterised main character. OK, Steins;Gate is better than Death Note really, but in turn Death Note doesn't have loads of pointless female characters just for the sake of selling merchandise... Hmm... Yeah. Come back to me later on that. Neither are fanservicey anyway. Unless I forgot what happened in them. I dunno.
Martian Successor Nadesico is generally alright but not outstanding in most respects, except it has the most inspired story I've ever seen in an anime, so... It's worth watching, even if other things are actually much better in most ways.
FLCL is weird and incoherent. It's also a beautifully thought out completely coherent story. It's a work of genius. It just takes several views to appreciate it or to not get a headache while watching.
Full Metal Panic: Fumoffu, is a slapstick cartoon about some military guy going to high school. It's hilarious.
And finally, Puella Magi Madoka Magica. Madoka is an experimental anime, in that it uses interesting or unusual animation techniques to take a story that otherwise might be fairly good, but nothing great, and turn it into something truly outstanding.
Overall, I recommend that you don't watch anime and watch old Cartoon Network shows instead. They're far superior. Unless it's Ghibli. Or Gainax. Those are far superior to most things.