First, start off in the Wii's Main Menu where all the various channels are displayed. In the lower left-hand corner is a round button labeled Wii. When you hover over it the tooltip is "Wii Options". Start by clicking that button.
Now you have two big options to choose from. Select the one on the right, Wii Settings.
In this next menu you'll start out with a tab at the top labeled Wii System Settings 1. There's also a series of three rounded rectangles numbered 1 through 3 at the bottom right-hand corner. You'll need to press a right-pointing arrow button that's on the right-hand center side of the screen. Now you're in the aptly named Wii System Settings 2 tab. Choose the Internet option.
At this point you only have three choices. Pick the middle one, Console Information. And there is you MAC Address. Its just a series of twelve hexidecimal characters where pairs of characters are seperated by dashes (unlike the colon seperators for the iPod Touch). You can safely ignore the LAN Adapter MAC Address so long as you're only interested in the Wi-Fi networking for the Wii. The first half of your MAC address should be "00-21-47".