Easiest way to do it....
Go on your Wii, go to the Shop Channel.
Get out your credit card, set up an account and buy 1000 points for $10.
Buy the Internet Channel with 500 points, and you can now browse the internet on your Wii.
But everywhere you need to type something, you have click on a virtual keyboard, it's a pain in the ass, but you can just have favorites, and go from there.
Edit: Favorites are only for web sites, you still need to type shit in with the Virtual Keyboard like sign ins, searches, etc.