Do I have to pay monthly for Apple Watch cellular?

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

But if you do not want to activate the cellular radio, you can use it as if it is a GPS only Apple Watch, and not pay anything.

Or if there is no competing reason to buy a Cellular, you can save money and just get a GPS only to start with.

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.

Apple Watch Family Plan still requires an iPhone. It is intended for dependent children or aging parents, where an adult uses their iPhone as the controller of things like placing restrictions on what children can do, turning off features during school hours etc...

https://support.apple.com/en-us/HT211768

https://support.apple.com/guide/watch/set-up-a-family-members-apple-watch-apd54d0a51fb/watchos

https://www.apple.com/newsroom/2020/09/apple-extends-the-apple-watch-experience-to-the-entire-family/