High school drama? It's gotta be Haruhi. The Melancholy of Haruhi Suzumiya, highly entertaining and with an awesome deadpan snarker narrator. Due to circumstances beyond his control, he is dragged into the odd girl Suzumiya Haruhi's hijinks, which lead him into events beyond his imagination.
If you want more regular slice of life high school comedy, Azumanga Daioh. It follows several girls around their high school life and adapting to their newest classmate, a brilliant elementary school student who skipped a bunch of grades. This show is beyond hysterically funny, with occasional moments of testosterone destroying cuteness.
If you want to step into another genre or see some of the classics, then some of the recommendations here will do you fine.
Classics:
Cowboy Bebop. If you haven't seen it, try it. Bounty hunters in space, some of the most memorable characters in anime, and one of the best scores the legendary Kanno Yoko has ever compiled.
TRIGUN. Another classic character in anime, Vash the Stampede doesn't want to cause devastation wherever he goes, but trouble just keeps on finding him. Or so it seems at first...
Samurai Champloo. Not sure if this falls under classics yet, but it's hard to classify. Not your typical samurai anime, it follows a young girl named Fuu as she searches for "the samurai who smells of sunflowers" with her troublemaking bodyguards Mugen and Jin. Weirdly meshes old school japan with modern culture, especially hip hop culture.
Action:
Sword of the Stranger. Epic animated samurai film. Amazing characters, animation, music, you name it. Brilliant way to spend 2 hours.
Gundam 00:
Very solid mecha series, best of the modern Gundam shows I've seen. This is a decent introduction to a military mecha show, so if that interests you, try it out.
Tengen Toppa Gurren Lagann:
A very very off the wall and awesome mecha show. This series runs on pure manliness (seriously, the mechs are powered by determination, and giving manly speeches is accompanied by power ups) and kicks reason to the curb. Emerging from their underground settlement, where all humans have been forced to live by the beastmen, Simon and Kamina join the fight to liberate earth from the evil Lord Genome.
Ghost in the Shell: Stand Alone Complex
The mysterious public safety section 9 acts beyond the jurisdiction of most. Specializing in cybercrimes, they pick up the trail of an old cold case called the Laughing Man incident, which sparks a series of events leading all the way up the halls of power. Excellent cyberpunk thriller series, beautiful animation, intricate plotlines, highly memorable characters. Only complaint is that maybe half the episodes follow the main plots, the rest are stand alones (but still pretty good).
Dark Action: (Lots of blood. Soooo much blood. And mature themes, dark trends, not for the faint of heart or those under 18)
Black Lagoon. Salaryman Okajima Rokuro gets kidnapped by pirates and is left for dead by his superiors rather than paying ransom. When he turns things around, he finds he enjoys the pirate life better than being a grovelling salaryman, and joins up with the crew of the Black Lagoon. (Seriously do not watch if under 18. Not porn, but bloody beyond belief and some rather terrible things happen to people. Also some pretty brutal action sequences. Interesting characters, interesting setting, and amazing action sequences.)
Hellsing Ultimate: The Hellsing organization has protected England for centuries against the depredations of the undead. But somehow the number of vampire attacks is going up, and using their own vampire, Alucard, they must hunt down the source of these vampires before England is engulfed in a tide of blood. (So much blood. Brutal violence. People getting eaten on screen, etc... Not for the faint of heart. But operatically awesome in its scope and actually makes vampires scary instead of sparkly. Also, watch this in the dub, it is one of the few where the dub exceeds the japanese VA's and they weren't too shabby to begin with.)