Alright, so, you're getting into anime, don't know what to watch. You've heard talk of these "classics", but you're not sure which ones are actually good. Don't worry, I've got a list as long as you can imagine filled with titles you might enjoy.
First off, classics. You can't go wrong with the original films that helped to propel anime abroad - Akira, Ghost in the Shell, Spirited Away. Spirited Away is from a later generation, but, it's still one of the biggest influences on Japanese animation enthusiasts. In terms of series, you will want to give the following a spin:
(EDIT - OH GOD I CAN'T BELIEVE I FORGOT MY FAVORITE CLASSIC SERIES) Irresponsible Captain Tylor
Tenchi Muyo! (The original 14 eps)
Hidamari Mason Ikkoku
Star Blazers (Space Battleship Yamamoto)
Galaxy Express 999
Anything made by Leiji Matsumoto, really.
And, lastly, Dominion Tank Police. There are other classics, but these are the basics that will introduce you to most genres.
In terms of newer media you might be interested in:
Mononoke (NOT PRINCESS, just Mononoke.)
PRINCESS Mononoke (Mononoke-hime)
Wallflower (Yamato Nadeshiko Shichihenge)
Tramps Like Us
Sayonara! Zetsubou Sensei!
Any of the Negima! OVA's made by SHAFT.
Anything by SHAFT and MADHOUSE, really.
Isekai no Seikishi Monogatari (The only Tenchi spin-off series worth watching)
Hana Yori Dango (The single most popular Shoujo in mainstream Japanese media)
Fruits Basket (The single most popular Shoujo in mainstream American media)
The XXXHOLiC movie - if you like that, check out the series, but only until season 2 of the anime is it any good.
Why aren't you watching Code Geass again? LELOUCHE DIED FOR YOUR SINS.
Gundam Wing - watch it for Triez. The most manly of men, the kind of man who remembers of the name of every soldier who died in battle under his command.
Do NOT pick up one of the uber-huge series like Naruto. The problem with those lie in the creation process - they crank out episodes for each chapter of the manga without giving the author pause, leaving huge gaps and filler made with little funding and poor quality animation. It will turn you into a lifeless husk.
Also, don't watch Boku no Pico, no matter what 4chan says.
No, seriously, don't watch it.
Stop, why are you Googling it?!
OH GOD NO
You also might want to check out Dirty Pair, movies by Studio Ghibli, and I must agree with another poster - Bacano! and Durarara! are both works of genius. Watch 'em, enjoy.