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.
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.