How do I update the apple ID on iCloud?

I changed the email address associated with my apple ID. However, my old email address is still showing in iCloud. It keeps asking for my password, but won't let me in because the ID is wrong. How can I update the apple ID to the show the correct email address in iCloud? I did update the email address in the users & groups under "system" on my mac, but that didn't solve the problem.

Thanks