At the moment there are no online games. However you can trade miis and send messages by adding other peoples friend code. To find out what your friends code is just go to the message boards on your wii then open up the address book and on the first page your code should be listed there. To send messages/miis both of you must add each others friend code.
As for online games there aren't any at the moment.
If you want to learn more about Wii Online etc I suggest you check out the Wii Online Section. I also suggest you read the forum rules.