As far as I'm aware, once you have purchased the Watch, you simply need to setup on your iPhone. When it comes to the Cellular connection, it will re-direct you to your providers portal. Here, you can log in and add the Apple Watch eSIM option to your current plan.
Please note that your Apple Watch contract has to be linked onto your iPhone provider contract and your provider must support eSIM connectivity.
If your iPhone contract is with Verizon, then you should just be able to go through and set up the Watch and add it on to your plan.
The Apple Watch is essentially like an Unlocked iPhone. It can be setup to work with any provider, however, for now only Verizon is supported. If you had to buy the Watch from Verizon to use it, then Apple wouldn't sell the Watch in the US as nobody would be able to use them (Seems like Verizon's Logic).
Hope this helped.