Do a full sync and backup of your d

Do a full sync and backup of your device with iTunes, making sure you're saving all your contacts, photos, etc. (especially if you have iTunes set to not sync something). To make sure you have a backup, you can click the button in iTunes that says "Back Up Now" (on the "Summary" page that's available when you plug your device into your computer). Or if you want to save data with iCloud instead, double-check to make sure iCloud is set up the way you want.

Make a list of your Cydia packages. You can write this down by hand, or use AppInfo to email yourself a list of your installed packages and sources so that you remember which ones to reinstall after restoring. (Technically you can use AptBackup, PkgBackup, or a similar tool to backup your tweaks, but this isn't a great idea if you're upgrading all the way from iOS 5 to iOS 6 (or iOS 6 to iOS 7) due to package incompatibilities. See this guide to Cydia backup tools.) You'll be able to reinstall your purchased packages on your upgraded device.

Restore with iTunes, which will update your device to the latest version of iOS available (or if you're interested in restoring to a version of iOS that isn't the latest version available, follow this guide to using SHSH blobs to upgrade and then skip to Step 4). If you're just doing a normal update to the latest iOS version with iTunes, you should click "Restore" instead of "Update", for best disk space results. If you need to preserve your unofficially unlocked or unlockable iPhone 4, 3GS, or 3G baseband (including 6.15.00), you need to use custom firmware instead of a normal iTunes restore (the guide to using SHSH blobs also preserves your baseband on older devices, so you're OK if you need to do both). If you have an officially-unlocked or otherwise IMEI-unlocked iPhone, you don't have to worry about preserving the baseband, but it's harmless to do so.

iTunes will ask if you want to load your backup onto your device or "set up as new". Most of the time, loading your backup will work just fine and is convenient. (Most of your jailbreak tweaks store settings in your iTunes backup, so loading a backup also reduces the amount of time it'll take you to configure your tweaks after reinstalling them.) But once in a while people end up with problems caused by bad settings from the old backup; if you use the backup option and end up with problems, you can always restore again and choose "set up as new" and then sync over your media and apps. (If the "restore from backup" option isn't available and you use Gevey SIM, see this thread for suggestions.)

Jailbreak and reinstall your Cydia packages using the method you selected in Step 2. To access your purchased Cydia Store packages, tap "Manage Account" on the Cydia homepage and sign in using the same Facebook or Google account you used when you originally purchased those packages. Some packages have their own independent purchasing systems, such as LockInfo and biteSMS, but those systems generally also provide a way to access previously-purchased packages after restoring and jailbreaking again.