Apple has made switching over your iPhone when you buy a new one very easy, but there's one other task you may need to undertake, especially if you're one of the many players of the popular World of Warcraft game out there (or Diablo III, or Starcraft II, or other Blizzard games): You'll need to switch your Blizzard mobile authenticator app over to your new device. Blizzard posted a reminder to do this over the weekend and their online guide to removing authenticators should help anyone who needs to transfer over with the process.
I just did it myself: You'll have to remove the authenticator from your previous phone device from your Blizzard account (by first confirming a few codes from it), and then add the new one (by entering the serial number and authentication code) once you've installed the app on your new device. It's not hard at all.
It is a little annoying that you should have to go to such lengths to protect your gaming account, but with hundreds of hours of gameplay and memories stored up in these games, it's a process you might as well undertake. If you grabbed a new phone this weekend and play WoW yourself, make sure to switch things over before you get rid of your old phone.