I'd like to reinforce the recommendation of Ghost in the Shell, but with a caveat; I only recommend the original manga/comic.
The first movie is just a mashed-together version of some of the manga's chapters without a lot of the world-building the manga has and with some really out of character lines from Motoko thrown in to be a mouthpiece for Oshii trying to sound deep and meaningful. "Trying" being the operative word. The major in general is not the same character at all in that movie, and the puppeteer story seriously loses a lot by having everything crammed together so closely.
I think Oshii was perhaps a very poor choice for this project, too. I feel like his style clashes with the material, aside from the material being too dense to truncate into an anime movie runtime in a satisfactory manner. He does a really great job with Patlabor 2, though.
On that note, the first two Patlabor movies come heavily recommended from me.
EDIT: Rant mode engaged.
I'm a bigtime fan of mecha in general (mostly humanoid robots, as far as anime is concerned) and Gundam in particular. The newer shows are utter slop, and 00 and Unicorn are the worst variety. The type that are inane and self-important and go out of their way to remove or destroy anything they might introduce that is remotely interesting. 00, as trite as it is, at least has SOME original thoughts in its head if not any good ones, which is more than I can say for Unicorn. If you really have any interest at all in the franchise, watch the original Mobile Suit Gundam. The movie trilogy is probably the easiest way to digest it for a newcomer. Beyond that, either go in production order or just use a friggin' dartboard. Or some combination thereof. I'd post my favorites, but I feel like they have little value without context and it's not like you could ask for a favorite Gundam in a thread of enthusiasts without hearing conflicting replies to begin with. And I'm sure those are just as garbage to others for different reasons anyway. Don't mean this post as an attack on anyone, but I felt like I wanted to say something. Opinions and all that.
Basically, there's a lot of good in the franchise but start at the beginning. For the uninitiated, the original shows (created by Tomino, the brain behind the original show) are Mobile Suit Gundam -> Zeta Gundam -> Gundam ZZ -> Char's Counterattack -> Gundam F91 -> Victory Gundam. Otherwise it's all mix-and-match. May god have mercy on your soul if you decide to venture into it.
is an interesting case also, I think. It's a 6-episode show about a teenage girl in a pilot academy (more or less) who goes from being unable to properly walk around in a trainer robot to saving the galaxy with the most powerful piece of technology mankind ever produces. And it's a believable journey is the crazy bit. It's also one of the few bits of sci-fi that tries to deal with time dilation (even if it doesn't get it quite right), and it has a pretty distinct 80s style to it with Haruhiko Mikimoto's art.
, a sequel of sorts is very good as well, but VERY different. If you expect more of the same as GunBuster you WILL be disappointed. If you give it a shot as its own entity though, you might have fun. It's a lot harder to describe, but the best way I can think of to word it is that an android girl strives to be a hero despite having no real training for it. Not dynamic-sounding I know, but probably best to watch its predecessor first anyway, especially since you won't appreciate the ending without it.
is a movie rec. Think F-Zero, Wacky Racers, with a slight dash of Fast and the Furious. It's VERY over-the-top, and very fun. It's all hand-drawn, but was released in 2009. It has a very unique soundtrack and a dinstinctive style with a lot of ink-style shading. Not very serious for the most part and pretty fun.
is a show that I feel I would be remiss as an anime viewer if I did not mention. It's a beautiful breakdown of the magical girl genre and a well-told story on its own merits. Most things I can think to say to advertise it would spoil some good surprises, so definitely go check out at least the first few episodes sometime.
is one that I must heartily endorse as well for the reasons Kelthaz
states. The first season, at the very least. R2 kind of...leaps off the rails after a bit, honestly, but it DOES come back around for an EXTREMELY strong ending.
There's...a LOT more I could recommend, but I'm tired and these are what came into my head. Sorta funny since MOST of what I like seems to be 80s and 90s but I didn't mention much. Maybe I'll post again later.