Well...you need the internet to update the firmware so that you can get SD card support (i think that was one of the issues), and also soon they'll release the Weather Forecast Channel (which your kids might not have much use for)
but honestly the only thing the Wii is good for online right now is the Miis!! exchanging friend codes with people so you can get their Mii's
this baby will work straight outta the box though, no worries. just plug it into your wall, your TV, put your sensor bar in a nice location either under or over your TV, pop in a game and youre ready to play.
VERY very simple, won't get much easier than the Wii, especially not with the PS3 or XBox360
You have some lucky kids, good gaming to you and yours ^_^