This is as far as I know from using my Wii, but I may not have everything exactly correct.
Internet access is technically free, but you have to be able to connect it to your current Internet connection. The OS is also not the best. It cannot play DVDs or CDs, though apparently there is a way to play .mp3 or .avi files off of an SD card.
It is backwards compatible with Gamecube games, and you HAVE to use a Gamecube controller in order to do so (though if you purchase games originally released on the Gamecube through the Virtual Console, then you I think you can use the classic controller – but I’m not positive.)
Basically, everything you are looking for is in an XBox 360 or Playstation 3 (which are built to be media centers.) The Wii has a few add-ons that are neat, but it is – first and foremost – a video game console. So if you are interesting in purchasing one, I would look through the gaming library to see if there is enough to persuade you.
The reason why I like the Wii is the huge library in the Virtual Console. You can purchase games from the NES through the Gamecube. That’s bloody amazing. There are plenty of Wii games that are worth buying the console for, especially Mario Galaxy and its sequel.