It's not automatic, but YES, you should buy a wireless router. You should have bought a router (even just a wired one) before now. Otherwise, you're PC's are not secure (don't trust software firewalls).
Once you get the router configured (make sure to change the router's admin password, change the SSID, Don't broadcast the SSID, enable WPA (and assign a key), then you'll need to go into the Wii's settings and manually enter the router's SSID, set the Wii for WPA and enter the key. At that point the Wii will be able to SECURELY access the router and you'll have Internet. More importantly, no one else will be able to get to the net through your connection, do illegal things and let you take the blame.