How much moving around you do, really depends on you. You don't have to get-up and move around (unless it's something like DDR or the upcoming WiiFit), but with some games, moving around
does help to make the experience much more immersive.
The Wii comes with
WiiSports - a collection of sports titles (baseball, bowling, boxing,
golf and tennis) intuitive and simple enough that just about anyone from any age group can pick-up and play. Also included is the Wii remote and a Nunchuck (which is an analog stick attachment). you will want to at least purchase an additional Wii-Mote and Nunchuck. If you own a GameCube, the Wii is 100% backwards compatible will all GameCube games and the GameCube controller can be used for the Wii Virtual Console. If not, you will want to also buy a "Classic Controller" for the Virtual Console. Based on the list of games your family enjoys, it looks like you will absolutely love downloading the classics from the NES, SNES, N64, Genesis (Mega Drive), TurboGrafx-16 and coming very soon: NeoGeo & TurboCD.
HERE is a list of games currently available for download in Europe for the Wii Virtual Console.