The iPhone will access the internet via either wi-fi or the 3G phone network only when you tell it to, by way of the Safari browser or one of the apps. As KaiHallarn says, it uses iTunes on your computer to search for operating system updates (by the way, if its a jailbroken or unlocked iPhone do not upgrade without finding out a solution to re-unlock it - updating will re-lock it to its original nework. The latest update, 3.1.3, has no unlock solution as most hackers are waiting for 4.0).
Using 3G will obviously use your network internet data allowance (I pay Orange £5 per month for unlimited data download). 80mb is a lot of data, unless you're downloading songs, videos and apps. Just surfing the web doesn't use much data, especially as a lot of websites are mobile friendly.
Using home wi-fi is just like using a computer on wi-fi. Downloading data will use your home internet allowance.
If you have the iPhone registered to one or more wi-fi networks it will always, when in range, automatically connect to them and use them instead of 3G.
You can see if the phone is connecting to the internet by the icon bar at the top of the screen. To the right of the network name you will see either 3G or the wi-fi symbol (a point with two radiating lines) to indicate how its connected to the internet. When it is actually accessing the internet (by whatever method) you will also see a circle of eight small lines which light up consecutively to show you data download rate. The slower they are lighting up, the slower the data speed.
Any app that finds your location will do it either by using the onboard GPS (which is free) or via the phone network's cell system (which is also free). Any app that shows you where you are on a map usually downloads the maps from the internet.
Try getting Orange Wi-Fi app which will search your location for Wi-Fi hotspots. Unfortunately it does this by accessing the internet.