How to create a new Apple ID on my iPhone only?

Click here to see how to 'Create an Apple ID' = create an Apple Account. --> https://support.apple.com/HT204316

The Apple ID will only be other devices when you intentionally sign into that account on those devices. Don't want it on the Mac then don't sign in on the Mac.

I am not sure about the Watch. I don't own one but I know it is pretty dependent upon the connected iPhone. You'll have to read up separately about that. It may "inherit" its Apple ID from the iPhone to which it is connected.

If you change you Apple ID on the phone and want to leave the old ID on the watch, see the article below. You will have to set up Family Sharing between the 2 Apple IDs.

Set up an Apple Watch for a family member - Apple Support

Setting up Family Sharing also allows you to share purchases between the IDs.

Family Sharing.   *

Family Sharing - Set up.  *