There are two ways to root your TF300. The first one is for a locked bootloader, and the second one is for if your bootloader is unlocked. :)
You must be on Build number 18.104.22.168 (Check at settings -> about tablet). If you have already updated to .26, than you must downgrade. The easiest method is this.
Once you are on .17, you can root your TF300 using the apk file in this thread. Just follow the steps, Now you have root! :)
If you want to update to .26 again and want to keep your root, you must download OTA RootKeeper. This will let you save your root and restore it after you have done the OTA update. Just open the app, follow the steps, and you're good to go = update. Warning: OTA RootKeeper will not work if you manually update your TF300!
You first need to flash a recovery. I recommend TeamWin's Recovery Project. ClockworkMod Recovery is of course also a great option.
Once you have either TWRP or CWM, you just have to flash the zip in this post.
Make sure to thank the makers of the apps & methods :)
Edited by Firemars, 26 May 2012 - 03:37 AM.