If you don't have a password for your wireless internet choose no WEP. If it still doesn't connect look at getting the Nintendo WiFi USB dealiemajig. It's for Nintendo DS but their website names it as something that works for the Wii too. If you don't want to try that or it still didn't work, wait for the Wii Lan adapter to come out. Last time I checked was a few weeks ago and it hadn't been released yet.
Edit: My Wii isn't online either, but it was on two separate occasions for a grand total of a minute and a half
