So recently I've been spending a lot of time watching anime series. For the longest time I was against it. Why? I really have no idea. Until college I had seen a little Gundam through my siblings and some Pokemon when my parents weren't home. I got to college and that quickly changed.
My room mate at my first college was appalled that I had never seen any anime. So I decided to humor him and watch one. I selected Gundam SEED because I was familiar with Gundam and SEED itself was touted as one of the best by many. I watched SEED and SEED Destiny in two weeks. I was hooked. I watched mostly Gundam at first, but expanded my horizons over time.
So here's a list of what I've seen in no particular order:
Gundam SEED Destiny
Fullmetal Alchemist Brotherhood (way better than the first series IMO)
Eureka Seven (Fantastic Series and I especially liked the music)
Kaze No Stigma
Aquarion (I really liked the mythology component and I've heard the second series is really good)
Eden of the East
Van Dread (at first I didn't like this series, but it quickly grew on me)
D. Grayman (Very good, but I wish they would dub the second half of the series)
Chrome Shelled Regios (Very good, but left a lot of questions at the end)
Soul Eater (I was rooting for Soul and Makah (sp) the whole way and even though it didn't turn out exactly how I would have liked, still excellent).
Darker than Black (The second series needs to be dubbed, but one of my favorites)
Full Metal Panic (both series, and the Fumoffu, which was hilarious)
There are a few that I have started and not yet finished. Some I probably won't ever, but may get back to.
Kenichi: The Mightiest Disciple
Also on my list to watch are:
Neon Genesis Evangelion
I've heard good things about all of the last three and hope that they are as good as others have suggested they are. I should note here that I really like to watch the show, enjoy the animation, without reading subtitles. As such I usually don't watch subtitled animes and only watched dubbed ones. I feel like reading takes away from the visual aspect that really makes an anime an anime. I know that a lot of people won't watch dubs and that this can be a touchy area for some, but having not liked anime for so long I feel that this is an okay compromise.
I highly recommend any of the animes I have listed above. Eureka Seven and Aquarion are probably my two favorite to date so I would definitely recommend those above all others. Most of the ones I listed above can be found on Netflix (currently) and some can be found on Hulu as well.
This list is not complete as I know that there are really good series out there that I still haven't found. I hope to add to it going forward.