Stupid Question: How does a Verizon upgrade work?

I’m due for my first upgrade with Verizon on September 11 of this year. I’m just wondering how that works.

Do I “sell” them my phone and they take that amount off of a new phone? If so, do they give you a decent amount or am I looking at getting $20 off of my next phone? Also, is the price of my new phone the “with 2-year contract” price since I’m re-upping or is it the full price?

My husband is visually impaired and his Droid Incredible’s 4-inch screen just isn’t cutting it. I want to get him the HTC One Max because it has a 5.9-inch screen. The “with 2-year contract” price is $300, but the full price is $600. I’m not paying $600 for a phone when he got this one for free with our contract.

So, if someone could explain this to me, that’d be great!