How can I connect my Wii to the internet to download things and play Nintendo wifi games

If you have a router or cat5 output near you, you can just plug an Ethernet (cat5/6) line from the output into your Wii. It should automatically connect after that.

Alternative: Wi-Fi

Go to settings and find network connection. Scan for access points and if you own WiFi your router should show up. Put your password in and tada! Internet!