This Wii's power button has three lights to determine it's current status; green obviously, is on. Yellow means that the Wii has "Wii Connect 24" turned on, which means it's often looking for updates and other sorts of content Nintendo sends to Wii owners, like famous game release announcements. The Red light means the Wii is truly powered off, and won't be constantly using it's internet capabilities.
And yes, there's a simple and easy way to to stop the Wii from doing all this internet business on it's own. When you turn on the Wii and go to the main screen with all the channels, in the lower-left hand corner, you'll see a button. Click it, and you'll go to another screen. Click whichever option isn't memory management (I forget the name, sorry), and scroll through the options until you find the tab "Wii Connect 24". Click it, and simply turn it off. Then the Wii should turn off fully and the power button's light turn to red.
Hope this helps.