Add my new iCloud account and remove old one

See Change your Apple ID https://support.apple.com/en-us/HT202667. 

And, here is What to do after you change your Apple ID or 

Password https://support.apple.com/en-us/HT204071  (supplementary to what the fist link says)

Post back if you have questions about what you can do if your Apple ID (1) is a third-party apple ID or (2) ends in icloud.com, me.com, or mac.com. 

If you have changed your ID or password, have you followed the steps in this article?

What to do after you change your Apple ID or password - Apple Support.