You'll need to give me more information
Do you have a Wireless Router. If not you might need to get the LAN Adapter for a Wired connection
Please tell us of your set-up
Such as
#1 Position of Internet
#2 Where the Wii will be stationed [In conjunction with #1]
Hmmm you might need to get the LAN Adapter. Although it won't reach far from your PC
Alternatively you can get a Wireless router. I do suggest this. It means other items such as Laptops can connect to it aswell as your Wii [Which has Built in WiFi]
Check the Nintendo website for a list of approved Routers [For the best results]
Yes u can without paying see a few thing but if u want the opera browser to surf the net u have to buy points. But to have friends and PM them u can use your connection without any fees.