Hmmm, for beginners?
I would perhaps suggest the following, although there's already some great suggestions previously:
Kanon -
'Key'/Kyoani-style drama. Kanon is excellent and still one of my favourites, it was also one of the first anime I actually bought the DVDs for. It also looks very good, especially in HD.
Kanon starts off a lil slow, but speeds up tremendously about half way through.
Air is older and shorter, but meant to be almost as good as Kanon so might be an easier watch?
I would recommend Kanon over Clannad, despite the critical acclaim, Clannad only really comes into its own when you combine it with After Story (the second season), and recommending something 50+ eps long at this stage might be daunting.
Bakemonogatari - Extremely unique style which'll make it different regardless of when you watch it; plus it mostly focuses on eastern folklore which most of us aren't too familiar with. It hits on more popular/net culture as well as 'otaku' culture (as an example one characters surname is Meme, and there are jokey discussions referencing things like DBZ which you don't need to know anime to get).
Has some good humour, some good theme music, fun characters and its one of my favourite series of the last few years.
Eden of the East - already mentioned, but great spy/thriller vibe going on, and the main characters have great chemistry.
Would also recommend for a newbie:
Higurashi - Slice of life, with a parallel universe and horror/psychological thriller vibe. Definately a good watch, and probably not something you need to have seen too much other stuff to appreciate. Season one and two, ignore the S3 OVAs.
Full Metal Panic - Great fun, that's all I can say. Also has a decent english dub which might be important. Guns, Action, Humour, Mecha. Lots too enjoy.
Elfen Lied - Action, Gore, More Gore, Adult Themes. Really good break-the-mould sort of show to demonstrate the genre has much more going for it than the more famous Shounen material. It won't be aired on TV anytime soon either (as confirmed by Adult Swim who *wanted to*)
Beck - Great slice of life drama about the trials and tribulations of a band forming, falling apart, reforming etc. Brilliant UK dub, I don't watch or even like many dubs but this one is great, probably the best I've seen (tying with Black Lagoon)
Black Lagoon - action show set in the indian ocean, a LOT of fun, and sometimes feels more like a western than eastern show. Great English dub, and I think its the one series I can think of the dub is possibly better than the original. The author actually went down on record and stated the series would be in english if he wasn't making it for an audience in Japan which says something.
Toradora! - Really good rom/com drama. Again one of my favourites, and not necessarily something you need to have seen much of previously. One of the best Rie Kugiyama 'Tsundere' shows.
When you've seen more shows and got used to some of the tropes, I'd recommend you checkout Genshiken and Lucky Star.
Also Spice and Wolf, but might be too slow for someone just starting to get into anime.
Shiki is a rather good, relatively dark Vampire drama, with a lot of moral shades of gray. It's only a few years old so it looks pretty good, and it's not the more normal vapid, pretty boy Eclipse style theme Vampires have been lumbered with recently, but the more traditional menacing form.