I believe there are three ways? Someone who knows better can come and fill in the details, but I've seen this thread's method, the dumbphone line method, and buying the new phone off-contract at full-price (or used off eBay).
This thread's method in particular involves Best Buy. Go to any new phone that you want and "Add to Package." This screen will pop up:
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Ignore the message in yellow and click "Check Eligibility." Next, it should come to a page prompting you to type in the number you're upgrading, the billing zip, last 4 digits of SSN attached to the account, and the billing passcode. Type those in and the next page should contain this if you have an upgrade available:
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Click that button and move on to this page:
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Click the "No Change" bubble, add the phone to your cart, and then check out as you normally would at any online store. Keep records of the site confirming NO CHANGE TO PLAN just in case. When the phone arrives 1 or 2 weeks later, go about removing the SIM card from the NEW phone/packaging and set it aside for now. Then, make sure your old phone's SIM card is the same size as the new phone's SIM card -- for instance, if you have a micro-SIM and you bought the One M8 or iPhone 5 you'll need to trim down to nano-SIM somehow, perhaps using a cutter via Amazon or med-heavy duty kitchen scissors, and sandpaper for thickness adjustment. Here's a video on the DIY matter:
https://www.youtube.com/watch?v=j6T1Mygucak
Once done, pop it in and turn on the new phone. Done? Do with the new SIM card what you will. I have a Bionic that I'm upgrading from, so I'm expecting an experience similar to mykl376 which may involve actually using the new SIM:
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Yes, exactly, only wait a week after it gets released. I don't know if it will let you choose the "Keep Current Plan" option as a new release, which was the issue at first when I tried to buy the One M8 the week it debuted. Waited a week and it was fine. Waited a week and a half and then the price even went down $50. It was great.