I had an extensive list somewhere. I'll edit it in if I find it.
EDIT: Didn't find the one I was thinking of, but I did find another, so here it is. Bear in mind I was a little younger when I made this, and wasn't such a seasoned poster.
For series these are all great:
Death Note - Obviously you would have heard of this by now. One of the most popular (and for a reason) short anime series out right now. Incredible story that keeps you hooked from start to finish, and very excited without any actual physical action happening on screen.
Naruto - Ninjas, one of my favorite long running anime (more than 50 episodes) as of now. Granted it's not the best thing since internet porn, but I am easily pleased and I personally love it, and later on the story gets rather twisted too. However nowadays it seems like its cool on forums not to like naruto and bleach, no clue why. Maybe if you like Manga, you could try it out in it's paperback form, as it's filler-less then.
Gundam Wing - Mecha anime (giant robots), its vintage now, was released in 91 i think, but the story is deep and the overall package is well put together, i still enjoy it to this day. Be warned though, it is heavily political, but never boring, i guarantee you that.
Bleach - Death Gods, demons and whatnot. If you like action and buckets of blood, then this one might be up your street. Again it's not going to win any awards, but it's fun.
Fullmetal Alchemist - Alchemy and a masterful story, definitely one to watch, short series. Recommended
Cowboy Bebop - Possibly one of the greatest anime series of all time, bounty hunters. The film is also very, very good. short series. Recommended
Dragonball Z -...I don't know. I loved it but I grew up with it, and while the action is great the episodes were stretched somewhat.
Ghost in the shell: stand alone complex - Cyberpunk, very deep storyline, very intricate, very interesting.
One Piece - Pirates, light hearted yet action packed, emotional, and hilarious at the same time, one of my favorite long running series.
Samurai Champloo - From the makers of cowboy bebop, amazing also, samurai, sword fighting and very well done.
Trigun - again an awesome anime, full of action and well developed characters, one of the best out there if the golden age (the 90s) i think, dystopian sci-fi. Recommended
Noir - In my eyes, a masterpiece. Assassins and amnesia, a great soundtrack, compelling story and a gunfight every episode, it actually had me hooked till the end. Recommended
Black Lagoon (I know i have put up a lot of action anime, but these are all great, and this is no exception. Mercenaries in south asia, gunfighting) Recommended
Code Geass
*More on the way.
And as for movies (without description, maybe I'll edit in some when I have time, but google is your friend)
Ghost in the shell 1&2
Jin-Roh: The Wolf Brigade
Grave of the fireflies
Perfect Blue
The Castle of Cagliostro
Cowboy Bebop: The Movie
Sword of The Stranger.
Tokyo Godfathers.