Basically you take the Wii home, get it out of the box, plug it into your TV and into the mains just like a DVD player, and then plug the 'sensor bar' into the Wii unit. You then place this same sensor either on top of or just underneath your television. The sensor bar is basically a long, slim grey piece of rectangular plastic with a cable running out to plug it into the Wii.
You then put some batteries into your 'Wii-mote' (Wii remote, included with the Wii console), and use it to turn the unit on. Of course, you're going to need a game to play. The Wii uses a disc format the same size and shape as a DVD; you just slot these into the unit. To get you started, Nintendo includes a game disc with the machine called 'Wii Sports'. This features Tennis, Baseball, Bowling, Golf and Boxing. To actually play the games, you'll have to go through a few menus. Just use the Wii-mote by simply pointing at a particular point on your TV screen and clicking the buttons (the Wii-mote is kind of half-way between a computer mouse and a TV remote - but so much more).
Tennis is much the same: swing the remote like you would a tennis racquet. In bowling you hold the remote as if it were a ball and 'throw' it (you don't actually let go of it!). Each different game makes subtly different use of the remote, and there are loads available to buy: from one called Zelda where you get to sword-fight using the remote; to Wii Play which features a number of fun games; and of course, Wii Fit...but that's another review.
I can testify that this is fun for everyone, from age 3 to 63 or older. Nintendo have been sorting out supply difficulties so it's generally now much easier to get your hands on one. If you are having difficulty, I would suggest checking out the Amazon Wii 'Customer Discussions' for helpful advice, just scroll down to under the reviews on this page. Good luck!