Here is a tutorial by nintendo you can check out:

http://us.wii.com/connect/

I will show you the steps I went through for a wireless connection.

Go to the Wii settings menu. Go right in menu until you find internet. Choose connection settings. Select either connection 1, 2, or 3. (Your wii can save the connection settings of 3 locations, so you can easily connect to your friends wi-fi if you go to their house etc..) Choose wireless if you are connecting via wi-fi. The wii has a built in wi-fi so you don't need to purchase any more hardware for the wii :) If you already have a router connected to your computer than select "find an access point". Select Ok to choose your router from a list of visible access points. If your router needs a password to connect than enter the password. (You will need to verify which encryption your router is using and select that encryption in the change security settings. i.e. my router is set using WEP) Save your connection and it will test it. I haven't used any of the other connection types, so for those I recommend looking at the link I gave you. Hope this helps.