I don't get why some people like subbed over dubbed. Unless the translations are totally different when comparing them, it really doesn't make a positive difference at all and it only makes it harder to enjoy the show by making you look at the bottom of the screen the whole time. Now, if this is a 4kids dub where they change EVERYTHING, then subbed is always better. Anyways...
Cowboy Bebop
This is one of the most critically acclaimed anime of all time, and for a good reason. The acting in the dub is spectacular, the writing is intelligent and mature, and the soundtrack is really great. It also had one of the most memorable endings I've ever seen in an anime... That's saying a lot. This show is basically a space western, but every episode is just so different from the last that it really stands out. It's only 26 episodes as well so you don't have to dedicate a lot of time to it.
Neon Genesis Evangelion
There are a few characters in the dub that sound awful (mostly the secondary characters that don't matter at all), but all of the main characters, especially by the end of the series, do a fantastic job. The series is basically a hyper-symbolic, post-apocalyptic mecha anime involving depressed teenagers. It's pretty confusing at times, but it all comes together in the end (especially the movie End of Evangelion). You can also watch the movie adaptations of the series which were made with recent animation technology (and thus look incredible), Evangelion 1.11 and 2.22. The movies branch away from the main series and have a different storyline but they're both very good. The original, though, is a classic and is highly praised by both western and eastern critics.
Fullmetal Alchemist: Brotherhood
Do
not watch the original: Brotherhood is basically a more accurate version of the original anime (as in, much, much closer to the manga) with better animation and story. It's a very new anime and the most recent volume of it just came out on DVD a couple of months ago, and the first 3 volumes are on Netflix to watch instantly. It's very, very good and the characters are very well-developed and thought out. There is at least one major plot twist in every episode so it always keeps you on your toes. The music is also very good as well.
Gundam SEED
This is the first anime I really watched from beginning to end and it was a landmark for its time. The animation is pretty outdated now and it looks cheap at parts, but the story is very well done and the drama only increases with each episode.
Gundam 00
This is another very recent anime. Contrary to the name, the only relation this has to SEED above is that the characters pilot giant robots. This version of the show, in the first season, has a slow-ish start but by the end it becomes so incredibly dramatic and intense that it's really worth watching to get there. Watch this in HD if you can because the animation is amazing at times. The second season, though, isn't quite as great as the first but both are definitely worth watching.
FLCL
From the studio that made Neon Genesis Evangelion comes the weirdest and most deep anime I've ever watched. At 6 episodes long it doesn't take a lot to get through, but you'll want to watch it at least twice to get the full picture of it all. The entire anime is basically one really, really drugged-up metaphor for puberty and growing up to become more mature. Sure, some things are just silly gags with no deep meaning, but if you pay really close attention you'll see some striking details. The soundtrack is also incredible on this one and I'd suggest getting it even if you don't watch the series. This series is also on Netflix in its entirety.
Sgt. Frog
Another series on Netflix is Sgt. Frog, basically a comedy which is sorta like Invader Zim in presence (alien comes to earth and absolutely fails at taking it over), but about cute frogs. It's absolutely laugh-out-loud hilarious at times and it makes a lot of pop culture references (the main character is obsessed with Gundam models, for example). Give this one a shot if you have the time, you won't regret it.
ANY FILMS BY STUDIO GHIBLI
Princess Mononoke, Spirited Away, Ponyo (to a lesser extent), Whisper of the Heart, My Neighbor Totoro, Howl's Moving Castle, Castle in the Sky, Nausicaa: Valley of the Wind, etc. Watch all of these; they're some of the best animated films ever made, including western animation.
Gurren Lagann
This one, I'm honestly a little mixed about. It's a very good show with great animation, characters, plot, and music (and it's made by the studio that made Eva and FLCL), but it has much too much fanservice to make it a really good choice. If you don't mind lots of random T&A and an entire episode dedicated to going to a bath house with little to no story development, then you'll like this one. The opening theme is great, as well as the second-half ending theme.
I hope this is a great list to get you started!
EDIT:
Code Geass
Made by the same studio as Gundam, Sgt. Frog and Bebop, Code Geass is a psychological drama that really makes you think. The main character has an ability to make anybody do what he wants them to do, but it only works once. This causes some complications, to say the least, as he tries to start a rebellion against the government. This has some T&A in it like Gurren Lagann, but not quite as much of it. It's actually very intelligent and has some really, really great plot twists in the end of the first season. I haven't seen the second season, but from what I hear it's not nearly as great as the first, like with Gundam 00.
