How do I add old Apple Account to my new iPhone?

Sign out of the one you are using and sign into the old one. Decide if you wish to keep the content and have it merge with the the content of the new account, or if you wish to only have the old account content on the device.

Read this document to learn how to sign out of iCloud on your iPhone, iPad, iPod touch, Apple TV, or Mac --> Sign out of iCloud on your devices - Apple Support

For an Apple mobile device:

1. Go to Settings, then tap your name. 

2. Scroll down to the bottom.

3. Tap Sign Out.

4. Enter your Apple ID password and tap Turn Off. (Read this if you If you forgot your Apple Account password - Apple Support ) 

5. Turn on the data that you want to keep a copy of on your device.

6. Tap Sign Out.

7. Tap Sign Out again to confirm that you want to sign out of iCloud.

Also of possible relevance:

Turn off Find My on your devices and items - Apple Support

Read this support article to learn how to sign in with your Apple ID on various kinds of devices --> Sign in to your Apple Account - Apple Support

For an Apple Mobile device:

1. Open the Settings app.

2. Tap Sign in to your [device].

3. Enter your Apple ID and password.

4. If prompted, enter the six-digit verification code sent to your trusted device or phone number and complete sign in.

If another Apple ID is signed in you may need to enter the password for that ID to sign out of that ID first.