If you have trouble getting Super One Click to work, download this:
http://theandroidsite.com/wp-content/uploads/2010/07/SGH-I897-Root-Update-1.zip
You can either download this from your browser of choice (with a download manager included within the app, of course) or you can download it on your computer.
Do not extract this .zip!
Place the file on the internal SD card, not in any folder.
On the phone itself, rename the file to "update"
Do not include the .zip extension in the name, as this will tend to screw your reinstall up, as exclamation_mark said above.
Turn off the phone. The next step is to reboot the phone into recovery mode.
There are numerous ways to do this, but I recommend trying the three button method first, as it's one of the most common and a useful thing to know, especially if your phone can't do it. I also recommend doing some reading over on XDA about potential bricks, rooting, and how to resolve them using this 3 button combo, or the download mode button combo, but that's at your discretion.
Anyways, to do this, hold both volume up and down buttons while holding the power button for about 10 seconds. The AT&T splash screen should appear, assuming you're rooting from a stock ROM. You can release the buttons after this.
This will bring you to a screen in which you will need to use the physical volume buttons to navigate up and down, and the power button to select.
Navigate down to "reinstall packages" and push the select (power) button.
What the phone will then do is install the update.zip file, and with it, root your phone.
When the phone reboots, it will look the exact same, but you will have a new app called Superuser Permissions. This app is a gateway app, which in turn, allows other apps root access.
Hope this method works for you!
If it does,
wait before flashing anything!
READ, READ, AND READ!
Do some research on what flashing a ROM is, all of the various ones, what they can do, what they can't, and the potential consequences of flashing. Read up on what kernels and baseband modems are, and which ones are out there. Be careful, and come to the forums if you need help!
Right now, I'm running Cognition 3.02 and love it! By far the most stable version of 2.2 yet, with an unmodified kernel and no OC at that! Mainly due to being based off of the Roger's official 2.2 release, I recommend this one as a good start.
[ROM] ★ Cognition v3.02 ★ Video Review ★ Smoothest Stock Experience! - xda-developers