How would I set up an Apple Watch series 3?

As there seems to be a lot of iPhone 6 users having setup issue. If you are one of them, then you would be advised to return your Apple Watch

Apple Holiday Return Policy (Purchase between: 15-Nov-2019 thru 25-Dec-2019, Return by 8-Jan-2020)

All purchases made before November 15, 2019 and after December 25, 2019, are subject to the Standard 14-day Return Policy. 

https://www.apple.com/shop/help/returns_refund

If you have a 6s, SE, 7 or newer, then follow the instructions ChrisJ4203 has given.

The same as any of the other series devices. It requires an iPhone 6s or higher, and the instructions are here, Set up your Apple Watch - Apple Support