Well what I would do is buy a router, because you can get wireless internet in your house, and most routers have an ethernet-out, which means you'd plug your modem into your router, then your PC ethernet cable (which is probably currently in your modem) into your router. then you type your routers address into your address bar on your comp and get some wireless security going. THEN you fire up wii, let it recognize the connection, and input the password you just created.
The thing is, with other options (LAN Adapter, USB connector) i think you have to manually input your comp and wii's IP addresses, DNS servers, and other techie stuff into the wii. If you do it by router it usually does it automatically. Just remember to power down your modem and router, connect router to modem, power on modem first, wait a minute(literally), then power on your router. this will get your router set up with your modem connection. sometimes people miss this step and have a signal but can't get online.