You download games through the Wii shop, but you have to pay for them. Obviously, you need to connect your Wii to the internet through the router first. You can either buy Wii points from most decent computer game type shops or just your credit card online.
I haven't looked for a while, but I don't think there's any way to download demos through the online stuff, unlike Xbox Live. You could see if any of the Wii magazines have demo discs so you can check them out or join an online games rental service such as
http://www.gamesmonster.com.
For brain type games, check out Big Brain Academy. Platform type games are pretty much covered by any of the Mario games, particularly Super Paper Mario and Super Mario Galaxy, both very entertaining.
Hope that helps