If I buy an iPhone with no internet, could I just use my own home Wi-Fi?

I asked my Dad if I could buy an Iphone (Meaning I buy it but he pays the bill minus $20 a month that I pay to help), and he said that I could but I couldnt get internet-- which of course I don't agree with. But my sister says that her friend doesn't pay for internet on her Iphone and just uses her own home Wi-Fi. Does that make sense?
Anyway, will I have to pay the extra amount of money to get Internet on my phone or could I just use my homes Wi-Fi?
I obviously won't be able to use internet anywhere else but that doesn't really bother me.

You MUST have a data plan with an iPhone. Unless you can find an unlocked iPhone for sale (which will probably be super expensive) you can't just use your own Wi-Fi.

Yes you can use your home Wi-Fi instead. I do this also but it kinda sucks that when you're on the road and you want to use internet to search sumthing you can't because of the fact that you don't have your own phone's internet. But overall it's a yes.

You can have an iPhone without a data plan and use your home Wi-Fi instead. But I doubt you can do so with carriers. You'd have to buy it unlocked and get a BYO phone plan (that doesn't involve data) and there you go.

Don't plans usually include data anyway (at least that's how it works in my country)?

You have to buy a data plan with an iPhone on any carrier. You may as well buy an iPod touch and install TextFree with voice.