How do I upgrade with the Upgrade Program?

You can do it in store or online. I did mine online as a new customer and picked up the device at the Apple Store.

 

We just upgraded my wife's 6s Plus to a 7 Plus last night at our local Apple Store. It was very easy, just walked in and told them she was eligible for an upgrade and after they verified that her phone was eligible, she got her new phone.

It might be a little different if you're trying to do the upgrade around launch time. I'm in the same boat myself, looking at either September or October for my upgrade from my 7 plus.

 

Upgrading is in store as you turn in the device then and there.

 

upgraded my wife from 6s plus to 7 plus. i reserved the 7 plus on the day it went on sale. waited outside the store for like 1.5 hours. went inside handed them 6s plus came out with 7 plus within 15 mins.