If I buy a watch with cellular capability so I need a contract with a provider or can I use my existing contract for my iPhone?

You have to be with a provider that supports the apple watch as the watch uses your phones number through an Esim;

Carriers that support the apple watch

The Apple Watch and the iPhone share the same phone number, hence your iPhone has to use a carrier that supports the Apple Watch. The fee in the U.S. is an extra $10 per month added to the iPhone cellular bill.