Apparently, the Wii Internet Channel (Opera browser) was free for a while. Neener neener.
I've only tried to web surf on the Wii without a keyboard, so I don't do it much. I might have to pick up a wireless keyboard and see if that makes the prospect viable, or at least more flavorful.
As to networking the HR20, I used this handy thread
at dbstalk. Short version is that there was a few clicks to update the HR20 and then a couple things to do on the PC. I've only done the PC stuff in Windows Media Player, but I think I'm going to try TVersity instead. I'm really just getting started, so I don't honestly know yet if this is going to be worth the effort. Another advantage of putting the HR20 on your network is that you can get (beta) DirecTV On Demand. The downloads are ridiculously slow, but there was some content available that I wouldn't have watched otherwise.