Carrier
this is kind of a silly question but if I buy the Apple Watch will it make my phone bill go up?
Apple Watch
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The Cellular Apple Watch shares the iPhone's phone number. This requires the iPhone cellular carrier to support the Apple Watch. The iPhone carrier charges a monthly fee for this service. In the U.S. the fee is $10 per month added to your iPhone bill.
iPhone Cellular Carriers that support the Apple Watch, which models, the radio frequencies each model uses, and the country where they offer service:
https://www.apple.com/watch/cellular
As mentioned by fieryjack you are not required to activate the cellular, and in that case a cellular Apple Watch will behave as it it was a GPS only Apple Watch (this would be the case if using your Apple Watch outside your carrier's service area, you wanted a different case that only comes in the cellular model, etc...)
The GPS only Apple Watch can do Everything the cellular model can do as long as your iPhone is nearby, where nearby is somewhere in the same 3-4 bedroom home (assuming typical wood frame construction). And of course when the iPhone is with you when you leave the house.