Step 1: Backup Your Device!
AT NO GIVEN TIME WILL A JAILBREAK PROCESS PUT YOUR DEVICE IN A 'BRICKED STATE' YOU CAN ALWAYS RESTORE TO UN-JAILBREAK.
If at any time your device becomes entirely unresponsive, it is safe to force shutdown ( hold the lock and home buttons together for ten seconds)
If your device is already jailbroken, there are a number of ways to backup your settings, tweaks, themes, etc. One I recommend is pkgbackup. While it costs $10, it will be worth it in the end (restoring everything manually is a pain in the ---) There are of course other free tools, but I recommend pkgbackup. If you had install0us before it went under, you likely still have appsync for your iOS version. If not, you'll have to grab it from another repo, as Hackulous is no longer in service. (A simple google search can turn up several repos which will provide appsync) I also recommend writing down your list of installed apps, tweaks, themes and such so in the case that your backup method fails, you don't have to start from scratch. After this, plug in to iTunes and sync your device. You are ready to update and jailbreak.
If you aren't already jailbroken, the backup process is simple, plug into iTunes, sync, and update.
After you feel that you have sufficiently backed up your device and are updated to iOS 6.0 - 6.1, you are ready for the next step.
As a side note, if you are worried about not liking the new iOS or compatibility issues, I recommend backing up your shsh blobs using tinyumbrella