Using new iPhone with current plan?

Alright, I’m not really sure about all this subsidized crap for cell phones. Here is my issue.

I have the current iPhone (8GB, bought it first day) and will have it for a full year June 29th. Now, come July 11th, will I be able to buy the new iPhone and just pop my current sim in it? (used to have verizon before the iPhone)

I don’t mind paying a full price for the phone if I can use it on my current plan. How does it work?