It costs nothing as long as you already have a connection to the internet. However you will need to download the internet browser. And doing so is free until I think June 30th. However downloading retro games will cost you, but you just can't point click and download. You must enter wii ponts or entire a credit card number.
But to answer your question. No. Surf the web all you want with it, as long as you have a connection to the internet, I assume you do since you made this post.