Nintendo has made it pretty hard to connect with other Wii owners through the Wii itself, but this hasn't stopped some intrepid web users from creating online trading posts for Wii codes, messages, game saves and Miis.
Here are a few of the better resources out there so far:
MiiPages - A country-by-country list of Wii owners looking for other Wii owners to connect with. Users can search for fellow gamers with similar games and interests and put in a friend request -- your Wii code is hidden until you accept a request. With over 1000 users already, this is shaping up to be a premiere Wii sharing community. Wii-play - An extremely large Wii forum and chat community that supports easy Wii code sharing and a clever system to redirect private messages to your friends' Wii consoles. The site is a little overwhelming at first and requires a little work to figure out all its features, but its worth it to find fellow Wii owners to connect with. ConnectMii - Basically just a forum to broadcast your Wii code to hundreds of strangers who have done the same. If you're friend list goals involve quantity and not quality, this is the site for you. Wii Save - A growing community devoted to sharing Wii save files through SD cards, complete with step-by-step upload and download instructions. The site is mostly Zelda saves for now, but hopefully we'll see a wider range of saves soon.
MiiPlaza - This site doesn't let you exchange Wii codes or data, but does let you share photos of your Mii Channel creations with the world. Kind of small at the moment, but sure to grow as more users try to give their Miis digital immortality.