Any Wii will work with any Wii game, though an (automatic) update may need to be installed from the game disc itself. In other words, a used Wii from 2006, or a Wii Mini, will be perfectly compatible with a game like Wii Fit, just the same as a brand new Wii would.
If you're looking for the absolute cheapest purchase, buying online would be the way to go if you wouldn't mind purchasing a used product. In stores you'll likely be charged $70 for a used Wii, but online you might find one for as low as $30 or so, still in perfectly good shape. For a brand new Wii, Google should give you a quick answer for that.
As for which to buy, the Wii Mini is a stripped down Wii. No online features, does not play the older Gamecube games, and other such cutbacks. It's essentially a bare-bones Wii meant to be as cheap as possible, but a regular Wii can be bought for dirt cheap as is... perhaps even cheaper. As such, you may be better off purchasing a regular ol' Wii.
It is important to note that the Wii and it's offshoots like the aforementioned Wii Mini should never be confused with the Wii U. The Wii U is an entirely new system with vastly improved technology inside it, and Wii U games will not be compatible with the older, now outdated Wii. I'm mentioning this because there's quite a bit of consumer confusion regarding the two... Nintendo did a right-poorly job distinguishing the two apart in their advertising, let me tell ya.