Generally, my top faves are listed here. Have rewatched most of those multiple times and they've got a special place in my heart :3 Additionally, here are some other titles, depending on interests and what people are in the mood for:
12 Kingdoms is an anime with great world building. Also, for new and old anime viewers alike, the first episode of The Tower of Druaga is a must see, since it parodies most cliches in anime and is hillarious.
Skip Beat as well as Koukou Debut are decent shoujo manga/anime titles. Ouran High School Host Club is a school anime with a reverse harem situation (multiple boys and just one girl) but is rather hilarious too. Other shoujo titles are The Wallflower and Lovely Complex and the recent but spectacular Kimi ni Todoke. Kare Kano is still my favorite though.
There is also Great Teacher Onizuka which is a classic school!title with focus on the teacher's side - it has a manga, anime and drama series of it (with a super young Oguri Shun <3)
XXXholic anime is a good Clamp title (with supernatural "I-see-ghosts" thrown in) - Watanuki is hilarious and Yuuko rather fascinating.
Red Garden (and the follow up OVA), Moribito: Guardian of the Spirit and Michiko to Hatchin are anime titles with kick ass heroines.
Gakuen Alice has cracky humor - other young!student anime titles are Card Captor Sakura (the classic main Magical!school girl title IMO) and Kodocha - also enjoyed Hyakko and K-On to an extent but it depends person to person. In my top favorite list linked above, Nodame Cantabile is absolutely awesome and insanely cracky.
For the mind fuck value, you have Evangelion, but I actually don't like it much (apart from the character Asuka) and tend to prefer stuff like Baccano and the currently airing Durarara. I even prefer Death Note, but only till halfway through the manga where it really should have ended instead of gone on. These days I'm more likely to watch Showtime's Dexter.
For samurai titles, there's Rurouni Kenshin which is kinda dated but still well regarded - a darker manga title alternative is Vagabond. People have also raved about Samurai Champloo.
Summer Wars is a recent awesome anime movie - Princess Mononoke is an older fave. Other movies: Howl's Moving Castle, Escaflowne: the Movie, Vampire Hunter D: Blood Lust (for the supernatural eyecandy) and Voices of a Distant Star
For sports titles (which are suprisingly crazy addictive) there are a boatload including Cross Game, Hikaru no Go, Initial D, Prince of Tennis, Slam Dunk, Eyeshield 21 and Hajime no Ippo
For mecha and giant robots, we have Full Metal Panic and the Gundam franchise (have only tried Seed, Wing and 00 though). Gurren Lagann is popular these days but left me with mixed feelings.
Unusual titles that don't quite fit anywhere are Tales of Agriculture, Genshiken, Sayonara Zetsubou Sensei, Soul Eater (shounen but kinda different) and Komatta Toki ni Wa Hoshi Kike which is almost a shounen-ai manga but not quite.
Darker than Black is a generally good/great shounen anime. The bigger shounen titles tend to be more formulaic but are worth marathoning when you have the chance, like Naruto, One Piece (the art takes a bit of getting used to) and Bleach. Fullmetal Alchemist (both the first anime and the second one, Brotherhood, and the manga) is fantastic. There's also D. Gray-Man which is kind of different too.
Yaoi/Shounen-ai anime (boys love as opposed to Shounen which is just anime targetted at male audiences) you have Junjou Romantica while a lot of the others tend to jump to the smut or are rather dated.
For manga titles of the slashy variety, there's Totally Captivated and Viewfinder, and crazy amounts of titles, which really depends person to person (seriously, shounen-ai/yaoi titles are very much hit and miss in a lot of cases).
Recent regular manga titles are The Breaker (a korean manhwa with fantastic art), Yanki-kun to Megane-chan, Otomen and Dogs: Bullets & Carnage
Something that feels like a cross between XXXholic, Natsume Yuujinshou and Mushishi is the manga Hyakki Yakoushou.
For far eastern dramas, I recommend: Coffee Prince, GTO, Nodame Cantabile, Kim Sam Soon, Style, Dal Ja's Spring, Gokusen, Powerful Opponents, Goong and period pieces like Seodong Yo, Jumong and The Legend. Again, its all about personal taste but as you go along, you'll probably get an idea intuitively of what would work.
That's about it - more of a scatter shot than anything else and I'm sure I've missed titles. Still, if anyone's been wanting to try something different, here are some options :D