If you changed your Apple ID by going to My Apple ID and clicking on the EDIT option to change the email address, then you will need to sign out of, and back into the following apps on all of your devices:
Settings>iTunes & App Store
Settings>Messages
Settings>FaceTime
If you were signed onto iCloud with that old ID and did not sign out before changing the ID, you need to do the following to get signed out of and back into iCloud under the new ID:
1. Go back to Manage your Apple ID and change your Apple ID back to the old email address. This is temporary. Do NOT verify it, do NOT change your password: Apple - My Apple ID
2. Go to Settings>iCloud, scroll to the bottom of the screen and click on Sign Out (or Delete if you are running an older iOS). When it asks for a password, enter your current Apple ID password and follow the prompts to sign out the iCloud account
3. Go back to Manage your Apple ID and sign in with the old email address and current password. EDIT the Apple ID back to the new ID. You may have to verify it again
4. Go back to Settings>iCloud and sign in with your new ID
Cheers,
GB