How to set up Wi-Fi so I can use internet on my iPhone?

So a few days ago, I got an iPhone 4. Whenever I go onto the internet on my phone, it gives me a 5-digit network and asks me for my Wi-Fi password. Even though I do have internet in my house, I'm not aware of the password. Because I'm not able to sign onto Wi-Fi on my phone, I've already used up half of my data for the month.
I called Verizon tech support and 2 hours and 7 people later, I'm still confused. The last man told me that the 5-digit network is probably one of my neighbor's, because he didn't see that my router was wireless. He said I need to buy a wireless router and then set up my Wi-Fi password so that I can use my iPhone to go on the internet.
Should I take his advice and buy a wireless router? Or is there something else I can do to get a Wi-Fi password?

You'll need a wireless router to get Wi-Fi on your phone, laptop, tablet, etc.