EDIT: I suppose I should explain that I've watched hundreds of anime series, OVAs, and movies. I stopped counting when my DVD/Blu-Ray collection reached four digits. It's not possible for me to pick one or two favorites; this IS my short list!
Berserk - The only thing better would be an animated version of Glen Cook's Black Company.
Cardcaptor Sakura - The finest magical girl series ever. My six year-old son LOVES it. Note: do NOT get the "Cardcaptors" English dub, which tried - with hamfisted ineptitude - to edit a heroine-centered show for ten year-old girls into an ensemble piece for eight year-old boys.
Fruits Basket - Possibly the finest shoujo romance series ever made.
Genshiken - A loving, slice-of-life tribute to anime nerds.
Gunparade March - High school romance juxtaposed with realistically brutal warfare. As realistic as one can get when there are mecha and aliens involved, anyway. DO NOT watch the sequel.
Infinite Ryvius - Lord of the Flies on a space battleship. Crushingly depressing, and certainly not for everyone.
KamiChu! - A grade school girl wakes up one morning as a goddess. Exploring Shintoism with a Miyazaki-like sense of wonder.
Mobile Suit Gundam 0080: War in the Pocket - A short, no-bull distillation of the qualities that made Universal Century Gundam great. No one gets out of this with a happy ending.
My Neighbor Totoro - If you still have any connection with your childhood sense of wonder, you will enjoy this movie. If you don't perhaps it will allow you to reconnect.
Outlaw Star - Just a whole lot of fun. Space pirates! Starships that fight with giant mecha-arms! Catgirl aliens! Guns that shoot literal magic bullets!
Planetes - A near-future hard science fiction series with a social conscience. IMO the best serious SF anime ever made. The lead writer, director, and composer later collaborated on the more popular (and melodramatic) Code Gaess.
Rurouni Kenshin: Trust & Betrayal - If all you know is the rather lighthearted TV series, this prequel tale of Kenshin's youth as a cold-eyed assassin will set you back on your heels. Makes me cry every time I watch it.
Someday's Dreamers - I have a weakness for "magic realism" (Google it). The world badly needs mahou josei series; this is the closest we have for now.
Super Dimension Fortress Macross - The Star Trek of anime. Yamato/Starblazers is the Star Wars.
Toradora! - A slow-building, character-driven romantic drama disguised inside a tsundere comedy.
Trigun - Sure, it doesn't always make a lot of sense. But it has *heart*. It's also one of the extremely rare English dubs I enjoyed more than the sub.
Zone of the Enders - I will cheerfully endure any scorn I receive for my love of this series. An alcoholic 40-something space trucker comes into possession of a giant robot with the AI personality of a high school girl, and tries to reunite his broken family in the face of interplanetary war.