Switching Sprint iPhone to Verizon iPhone?

I currently have a Sprint iPhone and have come to the realization that although Sprint is a great company, great prices and great plans, they're coverage and speeds are extremely weak in my area. So I'm switching to Verizon. I will be selling my current iPhone on eBay, and my question is…
How do I backup everything on this iPhone and retrieve it on the new one (which hasn't come in yet) so its like I never did anything but switch carriers? How do I get my computer to recognize the new iPhone as this one I have now? They are both iPhone 5's, if that helps at all.

Verizon can switch your stuff on your phone. And by the way you won't be happy with verizon. They don't have unlimited data and they charge way to much for service.

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