I'm going to add a bit more to this as I'm bored. Simply the answer is "No" virtually every game you buy will have single player modes etc. You're not going to turn it on and be faced with a screen telling you to connect to the internet or else. However to play online portions of the games you buy (e.g. Smash Bro Brawl over the internet with other people plxplxplx be an option), to use the web browser Opera, weather channels, the planned messenger (I heard) and WiiConnect24 the always on download service as well as the Virtual Console in order to download the games you WILL need the internet.
Here's what you probably read, to play on the internet you either need a wireless router which the Wii will connect to through WiFi (wireless internet, seen a lot with newer laptops) or you will need an adaptor allowing you to plug your broadband line into your Wii like a computer.
Another alternative that may arise I think would be something like the USB adapter for the Nintendo DS where you plug it into your PC and the DS connects to the net through your computer wirelessly.
Here's a solution to your "My dad uses the net all the time" problem, buy a wireless router, it's that simple. You can get them remarkably cheap nowadays and with it your dad can still have his wire plugged into his computer so no need to buy a wireless card etc for it and you can connect to it wirelessly with your Wii. That my friend is a Win Win situation.