Get iTunes 11 up and running in a computer. Plug the old phone and tell it to sync up everything: contacts, dates, pictures, videos, etc. and then do a backup. When complete, unplug and plug the new i5. It will be detected as such and iTunes will ask whether you want it that way or if you want to Restore from the backup you've just done. Restore and all the info will be migrated over.
Note that not all the apps will move and may require manual reinstallation.Likewise with passwords.
If your iPhone 5 is already setup/activated, you will not be given the option of restoring it from your previous iPhone backup. If this is the case, and I suspect it is, you need to start over. On your iPhone 5: Settings>General>Reset>Erase All Content & Settings. Once done, follow the directions here:
Assuming you're using Itunes 11 I'd recommend this as it's generally the easiest way to do it.
If you don't have the side bar, hold down CTRL and press S.
It'll pull up a sidebar on the left side of itunes.
Under devices you should see your device there.
Right click it, and select backup.
Now remember, your new iphone won't see your old iphones backups if your old iphone is on 6.1 and your new one is on 6.0.1/6.0.2
Once the backup is complete. You can plug in your iphone 5, find it in the same device list and select " Restore Backup"
The rest of it is pretty simple/automated.
I migrated all the stuff from my boss' old i4 w/iOS 4.2.2 to a new, activated and working i5 w/iOS 6.0.1 yesterday following the procedure I detailed above on a rMBP with 10.8.2 and iTunes 11.0.1. Only some apps did not carry over.
Afterwards I upgraded it to iOS 6.1 using iTunes, again without any issues.