To get your Wii online you will first need to have a high speed internet connection.
For a wireless connection you will need to have a wireless network access point, aka a wireless hub. The Wii works with most standard wireless hubs. If you don’t already have wireless access set up in your home, you can read a simple description of how to do so here, or a much more detailed description here.
For a wired connection you will need an Ethernet adapter. I used Nyko’s Net Connect. Plug it into one of the Wii’s USB ports. The USB ports are the two small, rectangular slots in the back of the Wii. You will also need an Ethernet cable running from either your modem or from an Ethernet broadband router attached to your modem.