Q: How do I change the Apple ID on my iPhone or/and add it to my new Apple account?

Settings > ITunes & App store.

Tap AppleID, sign out and sign back in.

Settings > Messages.

Turn off iMessages then turn it back on.

Settings > iCloud.

Delete the account then add the new account.

I cannot find delete account under iCloud. In any case before I can do anything it asks me for my password on my old id.

oslerman wrote:

I cannot find delete account under iCloud.

It is now called Sign Out at the very bottom.

In any case before I can do anything it asks me for my password on my old id.

You updated you AppleID with new email address?

-> Apple ID: What to do after you change your Apple ID - Apple Support