I've actually switched from T-Mobile to AT&T when the iPhone 4 came out, as well as switched from AT&T to Verizon when the iPhone 5 came as well so hopefully this info helps:
When I went from T-Mobile to AT&T, I actually pre-ordered the device on Apple's website for an in-store pick-up, which the Apple staff did everything for me in-store during the release.
When I switched from AT&T to Verizon for the iPhone 5 release, I was already out of contract with AT&T, and decided to have it mailed to me instead as opposed to going in store, and all I had to do was go to Verizon's website to pre-order the 5 as if I am a new customer. During my order, they will ask you if you would like a new number, or if you'd like to transfer your number to their carrier (which you should choose). Nothing will happen until you receive your new device.
Once I got my 5 in the mail, I simply activated it thru iTunes and my iPhone 4 (from AT&T) instantly stopped working, and my iPhone 5 became active right away on Verizon's network. I received a letter/bill from AT&T the following month indicating my cancellation of their service, as well as any fees because it cancelled in the middle of a billing cycle.
OR if you wanted to switch over now:
Yes, you can actually use your Verizon iPhone 5s with AT&T. It came factory unlocked and I've actually tried it out myself with my friend's AT&T sim card and everything worked fine.
You won't have to go through Verizon to cancel - you can go straight to AT&T and they can port your account there when you sign up with them, and they'll automatically cancel Verizon for you.
Just a heads up - when you do get your iPhone 6, you may or may not still experience activation issues. Since a lot of people will get their devices the same time as you, Apple's server may be bombarded with people activating their devices, which is why I suggest going through my earlier post so that it's at least all done at once.
Hopefully that helps!
I'm planning on switching to AT&T from Verizon as soon as the iPhone 6 is released. Should I switch carriers a week prior to release date or on pre-order day? What should I do?
I want to ensure that I get the new phone on time and not by October/November.