I already changed it on my MacBook Pro, but it keeps asking me to put the password in for old Apple ID.
3 Answers
Go to iCloud and sign out. It will ask what to do with the data on the phone, and unless you don't have recent backups or worry that some changes haven't synced to the cloud yet - choose to erase the local data.
When the old account is cleaned up, sign back in with the correct password.
Similarly, if you have the old account used to download apps or sync with email services, you should delete those as well and then start using the new Apple ID for those or choose to sign in again to keep using the old services and apps with your previous Apple ID.
Everything you already got from the Apple Store is registered to your old ID, & cannot be transferred.
The only way to transition the machine to the new ID is to re-purchase everything under the new ID...
... or by setting up the two accounts under Family Sharing you would be able to use the same apps under both accounts.
Under Settings, go to iTunes & App Store, then tap your ID at the top. In the menu that pops up, you can choose to log out. Then you can enter another Apple ID and password.
On the Settings screen you'll also find an entry named iCloud. This is the account where your calendar, contacts, notes,... are stored. It's separate from your Apple ID, under which your app, music and ebook purchases in the iTunes store are registered. Of course, both can use the same email address.