It's the game the board game that was shown in Pi, A Beautiful Mind, and Hikaru no Go.

Do any of you play this (besides Itchy and Houdoken)? My handle on KGS is Pillow Face in case anyone here does.

This is a good introduction for people wanting to test the waters:
http://playgo.to/interactive/

If you are not discouraged by that, then:

Eye Shape: http://senseis.xmp.net/?EyeShape

Memorize by heart, and apply them in...

Tsumego: http://senseis.xmp.net/?LifeAndDeath

The best way to get good at tsumego is to do as many problems as you can, and repeat the ones you can't solve instantaneously until you can look at them and know the solution.

You can find 300 problems to get started here: http://senseis.xmp.net/?BeginnerExercises. They have a PDF of the problems and solutions if you want to print them out. The answers on the site provide in-depth analysis, while the PDF solutions just show the correct answer.

Shape: http://senseis.xmp.net/?Shape

This is good to know in general play to avoid tragic moves. In all that void between the big moves and tsumego, it helps to know what (not) to do.

Joseki: http://senseis.xmp.net/?Joseki

Read that and then go to http://eidogo.com/. Memorize the sequences and try to apply them in your games. I'm trying to do this, and it is
not really that fun, but neither is not improving, heh. Try to memorize all the variations that stem from Move 1: 3, Move 2: f, as those are the easiest to get under your belt and apply.

You might notice patterns/similarities among the variations of the joseki end results, which I think is helpful in memorizing them. Playing proper opening moves will help stack the cards in your favor later on in a game. Even if you don't understand why you're doing what you're doing until after you've memorized them, used them, and reached higher levels, you wouldn't get to those higher levels otherwise.

Here are the go resources that I keep coming back to:
http://senseis.xmp.net/?StartingPoints - more specifically, the Pages for Beginners link on that page. The commentators often trail off on a lot of exceptions, which are useful, but don't let it distract you from the main points.
http://gobase.org - I haven't actually looked at this, but I hear they have some good problems?
http://goproblems.org - searchable database of all kinds of go problems. I don't really care for the interface, but maybe you can find a way to make it work for you.

Finally...

http://gokgs.com - You can play games online there. Read this (http://senseis.xmp.net/?WelcomeToKGS) and this (http://senseis.xmp.net/?KGSTips) before you play.

There's a saying in go, lose your first 100 games as quickly as you can. Really, the best way to get over that initial steep learning curve is to play as many games as you can and have someone better than you review them with you.