I have an iPhone 4S does it cost me my Wi-Fi to go on at home?

I have an iPhone 4S and it has 1gb per month of Wi-Fi (3G) and I'm wondering when I come home and use my Internet Wi-Fi does my 3G or 1gb get used up?

Nope as you are using your home internet

No. But make sure you are in fact using the Wi-Fi connection and your phone doesn't just have the Wi-Fi antenna turned on.

No it don't cost anything but to make sure your phone is actually using Wi-Fi and not you 3g network, Just turn off the mobile network in the settings while you are in the house.
This will 100% guarantee that your phone is only using the Wi-Fi and also save a little battery on your device also.

No when your using your Wi-Fi your using the amount of data you get per month from the router

Your 3g data is for use when you are out and about, and can't reach your home Wi-Fi, or a friend's Wi-Fi connection. Your 1gb of Wi-Fi use from the network is for Wi-Fi hotspots, such as those found in pubs, cafes, etc. If you use your home Wi-Fi, or a friend's Wi-Fi, this usage doesn't come from either of your network allowances; as it is a separate usage, and costs you nothing, in terms of your network bill.