Not sure about exactly when the contract extends but I'm guessing it's when you activate the phone. I was nervous too the first time I bought online from VZW but really, it couldn't be easier. You don't have to go into the store at all. I don't remember the exact procedure but I think you basically dial *228 option 1 and it programs the phone OTA. There may be a few additional steps but it's simple. You can always call customer service if you want them to walk through it with you but literally, you should be able to have it all activated within a very short time of getting it. Again, no store trips necessary.
OK, here's the activation url:
Activate Your Phone Looks like you need the new phone, the ESN/MEID, the order number (of the online order you placed), last 4 digits of the social security number on the account. Enter the info, then do the OTA programming, and you should be good to go.