The console package includes the system itself (including the sensor bar and video cables), 1 controller, 1 nunchuck (the joystick thing that plugs into the controller for 2-handed stuff), and Wii Sports.
If you’re going to be doing 2-player stuff, a second controller is a must-have. Also, the controller doesn’t come with another nunchuck – you’ll have to buy that separately.
Also, don’t bother buying a memory card. You can save your progress to the Wii itself, so it’s not a crucial accessory like it was for the Gamecube.
Other than that, the must haves are mainly just various games.
The balance board is sold with Wii Fit, which is the game you were probably playing with it. That’s the one that has all the health and exercise stuff. There are some other games that support the balance board (such as We Ski), but you’ll have to buy those separately.
Super Smash Bros. Brawl is easy to learn and pretty fun. It’s essentially a fighting game (like Street Fighter or Mortal Kombat), but the violence is cartoony in nature and opponents are defeated by knocking them off the board rather than anything more violent.
Mario Kart Wii is another fun one. It’s a cartoony version of a racing game. Instead of cars, it’s go-karts, and the drivers are all classic Nintendo characters.
Zelda: Twilight Princess is a beautiful game with a lot to do and explore, but that might be a bit too in-depth and complicated.
Lego Star Wars: The Complete Saga seems to be pretty popular. I’ve only played an earlier Lego Star Wars game (for the Gamecube and only covering Ep1-3), so I’m not able to vouch for how the Wii version plays (especially with regard to the Wii’s unique control scheme). However, the earlier one I played was pretty light-hearted and fun, and the game was very forgiving – unlimited lives and such.
It’s pretty straight-forward. There’s 3 things that plug into the back of the Wii: The power cord, the sensor bar, and the video cables, and each has a different shaped connector. Once you’ve plugged those in, the sensor bar goes on the TV, the other end of the power cord goes in an outlet, and the video cables plug into the back of your TV the same way that a DVD player would.
The default cables that come with the Wii require an available component input on your TV. That’s the one with the three holes, one (usually yellow) will be labeled video, and you’ll have a red and white pair for the audio.