What's a good series from someone new to Anime?
As alpharogue said, it depends very mutch on your liking... I'd suggest a couple of series myself (mainly stuff more recent):
Cowboy Bebop : this 26 eps series is a nice introduction to common themes in anime, from mechas to dark and gloomy stories, passing by some defenitively trippy stuff. It also has a very unnusual jazz-inspired soundtrack. This is a anime by SUNRISE, famous for their mecha series.
FLCL : This is not something I'd usually recommend to someone without some anime knowledge, but I'm tring to go picking only really good stuff... This short 6 OAV anime borderlines on experimental animation and has a wonderfull rock soundtrack as well, by a band called The Pillows. This one is by GAINAX, name you'll hear more than often in anime circles - they did Evangelion as well, and a few dozen more classics.
Serial Experiments Lain : This is another brainer title. If you're into real, Gibson-esque cyberspace concepts it's the new mark. It's also very artsy/rave like, and the soundtrack here is heavy, lots of dance music and experimental-type tunes.
Now for some movies:
Perfect Blue - This is a wonderfull animated thriller. The blurb says that "If Walt Disney met Alfred Hitchcock, this would be the movie thew would have done"... what's somewhat correct, and displays the prowness of japanese animation in making scary titles actually work. This is a work that carrys the name of Katsuhiro Otomo, that is also involved in stuff like Ghost in the Shell and Blood : The last vampire, all good movies that easily please both american and japanese audiences.
Castle of Caliagostro : I just picked this one because i'd be accused of not giving old-timers a shot. This is a famous movie by Studio Ghibli, based on a even more famous series called "Lupin the Third". Amazing animation for it's age, quite funny, and Steven Spielberg once claimed the first few minutes to be the best example of a good car-chase ever made.
Princess Mononoke : Another Ghibli movie, this one was a huge hit in Japan, and failed to some extent in America dispite their all-star cast of dubbing actors. It is a quite beautifull, somewhat ecologic tale.
Spirited Away : Heck, it just got an Oscar nomination(sic?) as Best Long Feature Animation! It can't be that bad