Will Apple activate my new phone or will I have to go to Sprint?

If you go to an Apple store they will activate it. If you buy it online and have it shipped to you (or you use curbside pickup) you will have to activate it, but you can do that just by moving the SIM card from your old phone to your new phone. Or call Sprint and have them activate the eSIM in the phone.

You will have to have your new carrier activate your SIM card. Apple does not activate carrier service. Once you get your new device simply call the carrier you want and have them activate your SIM card.

You can remove the sim from your current phone and put it into your new one.