OK I must be dumb today, but I can't figure out how to do this and Google isn't helping me either - can you describe, or point me to a link, on how to set this up? Thanks!
If you want to use this for your App Store account, go into Settings then "iTunes & App Store", and sign out of the previous account. Log in using then different (new) Apple ID.
If you want to use this for your iCloud account, go into your iDevice's Settings, then iCloud, then "Delete Account". It will delete the iCloud data on your phone, but it should still be available on iCloud servers, and any other device hooked up to that account. I haven't looked into how to transfer data from one iCloud into another.
This goes for Game Center as well. You can use a different one for your mean, nasty, trash talking Gamer persona.
Apple doesn't always make this clear, and I see a lot of people confused about this, but this is an option.
Another login that can use a different Apple ID is Find My iDevice.