I managed to get my Wii online. If you're having problems (error code: 51330) connecting it to your wireless router and you've already properly set the SSID, encryption type (WEP, WPA, or WPA2), and the password, remember to set your router to 802.11b only or mixed mode (802.11b/g).
I found out that the firmware in most Nintendo Wii's only support 802.11b out of the box. When you successfully connect to the net, the Wii will prompt for a firmware upgrade and that will enable 802.11g. So after the upgrade, you can set your router back to 802.11g only and you should be running fine.