Too buy an iPhone watch, does that start my contract over

The Cellular + GPS Apple Watch has an eSIM and shares the iPhone's phone number, and the carriers that support the Apple Watch charge a monthly fee. In the U.S. the fee is $10 per month. Here is a list of carriers that support the Apple Watch:

https://www.apple.com/watch/cellular

as in you just add a fee on a monthly bases, which you can cancel. It is not a new contract, and it does not directly affect the iPhone contract.

If you purchased your Apple Watch from a cellular carrier (which do not not need to do), and you are paying on an installment plan, then you will have a period of time when you must pay off the Apple Watch, but the cellular service is not a separate contract from your iPhone cellular plan.

The GPS only Apple Watch can do Everything the Cellular model can do AS LONG AS your iPhone is nearby, where nearby is generally somewhere in the same moderately sized home. The Cellular model allows you to leave your iPhone at home while you go for a walk, run, bike ride.

https://support.apple.com/en-us/HT205547

Your question isn't very clear, there is no such thing as a Phone watch and what contract are your referring to?