Any Studio Ghibli film is good.
The two animes that I'd really recommend are Cowboy Bebop or Samurai Champloo.
They're both made by the same guy (Shinichiro Watanabe) so they've got the same vibe throughout them. Plus Steven Blum is the creator's muse, so he's in both series (in english dubs, which I'd recommend for a newcomer) playing a main character, which is good, because he's got an awesome voice. XD
Cowboy Bebop is basically a Western Sci-fi, with lots of great bluegrass, soul and rock in the soundtrack. It follows Spike Spiegal (Steve Blum), Jet Black, Faye Valentine, Edward and Ein, the dog. They just go around the solar system collecting bounties (trying to, at least) and getting into some awesome fights along the way. It's got an overarching backstory for Spike, but other than that the series is pretty non linear, since you can pretty much watch any episodes in any order, which is also great for newcomers to anime.
As for Samurai Champloo, it's a similar concoction of two adverse elements, this time with fuedal Japan and hip-hop, which, dispite my initial skepticism, turns out to be an excellent combination. It follows a similar bunch of misfits, this time including Mugen (Steve Blum), Jin and Fuu. Like Cowboy Bebop, it follows a similar path of nonlinearity, though with stronger ties to the overarching quest that Fuu is on, to find a samurai who 'smells of sunflowers'. It's also the reason why the two master swordsmen, Mugen and Jin, are accompanying Fuu, as they're both roped into it by their debt to her.
Of the two, my favorite is Samurai Champloo, simply because I saw it first and enjoyed it more, tho Cowboy Bebop was amazing for me as well.
I'm also gonna go out on a limb here and say that you have a genuine option to watch the series in english dubs, since in my opinion, they're some of the bed dubs around.
Anyway, I dunno what I was really trying to do with this massive description, I hope it helps you out tho, since both of these series really deserve to be seen.
*EDIT* I've also gotta say that the action in both aeries is fucking awesome, soif you're wanting for some good gun action or swordfighting,then you pretty much can't go wrong with Cowboy Bebop and Samurai Champloo respectively.