What steps should I take in preparation for upgrading to Mountain Lion?
According to The Tech Scoop and Lifehacker, there are a number of things you should do before upgrading.


Make sure your computer will actually run Mountain Lion. Here is a list of all the Macs that will not support Mountain Lion.
Run Software Update. Go to the Apple menu > Software Update and make sure you have all the current updates installed. Make sure that you are at least running Snow Leopard 10.6.8 (as required on Apple's website).
Make sure you have enough space on your drive. Mountain Lion requires 8GB of space, so make sure you at least have that amount free (preferably more though). If you don't have the space required, you can use a disk visualizer such as DaisyDisk ($9.99) or Disk Inventory X (free) or use AppCleaner to remove old, unused applications. More detailed information on that is available from Lifehacker
Check  that all your applications wil run on Mountain Lion. RoaringApps is the best place to check. It has a status indicator, and a feature list of what feature the application supports
Repair disk permissions and disk. If you are running Snow Leopard, start from the install disc and open Disk Utility. If you are running Lion, shut down and restart while holding Command ⌘ + R (restart into the Recovery partition). In Disk Utility, click on your main hard drive and then click "Repair Permissions". When that's done, click "Repair Disk".
Thanks to cksum for commenting that repairing permissions before an install is not necessary. "You don't need to repair disk permissions prior to installing Mac OS X over a previously-installed OS. The installer will do this automatically." (Apple KB doc about repairing permissions)  
Backup your computer. This is the most important step. If something goes wrong while installing Mountain Lion, you want to have a backup to go back to. Also, if for whatever reason you want to revert to Lion, backing up your computer will make it a million times easier to revert. The Tech Scoop recommends using Disk Utility to backup your computer. Lifehacker recommends using either Time Machine or Carbon Copy Cloner to backup your computer. My preference is Carbon Copy Cloner, freeware software for cloning a hard drive. If you want to use the other options, check out the links above. Open Carbon Copy Cloner, select the source and the target and click "Clone". Wait for it to backup your computer...


If you've completed all the above steps, your Mac is ready for Mountain Lion! Go to the App Store and install Mountain Lion.