If you are "not a customer of Verizon", then you would need to become a customer of Verizon by opening an account with Verizon. Go to a Verizon corporate store.
Or are you trying to activate the device with another carrier? If so, what carrier and what country?
... I know that I'm supposed to activate the phone so I can use Verizon's service, but I'm having a bit of trouble with that: I've dialed *228 and hit 1 when prompted (to activate iPhone). Then, when I dialed my phone number Who is the carrier for this phone number? Do you have cell service currently with another carrier? You can only use Verizon's service if you are a Verizon customer....
... I'm not a customer of Verizon, so I don't have any account numbers or anything like that to work with. If you are not a Verizon customer, then you can't use Verizon's service. When you try to activate the phone with a number that is not registered with Verizon, it appears you are being sent to the "payment center" to pay for service before they will activate the phone.
If I were you, I'd take the phone to a corporate store, and see what would need to be done to get it activated - it will involve setting up a Verizon account, either prepaid or post paid.
Do you currently have cell service with another carrier?