If you Dad is paying for the Apple Watch, then it is paired to his iPhone.
To transfer the Apple Watch to your iPhone, your Dad has to discontinue his AT&T Apple Watch service, then he has to prepare the Apple Watch to gift it to you using the above instructions I provided above.
At that point the Apple Watch can be paired to another iPhone, and that iPhone (since you appear to be in the U.S.) can be using Verizon, AT&T, T-Mobile or Sprint.
These carriers may requires that you use a limited number of iPhone plans to pair an Apple Watch to your iPhone. As you currently use T-Mobile you could ask them what kind of plan you need to use in order to pair an Apple Watch.
If you want to move your iPhone to AT&T, then it would be a new account for your iPhone, or if your Dad has a family plan, perhaps added to his plan as a new line, then you can add the Apple Watch to your iPhone.