Cost of living in Stuttgart for a family

Whilst waiting for members to reply try www.numbeo.com for costs. Also read other posts here on the forum for information.

This depends largely on your expectation and spending habits.
The German average income for a family of four is somewhere between €3000 and €4000 per month, but some survive with far less while others spend a lot more and sill feel miserable. As a foreigner, especially in the beginning or if you prefer exotic foods or habits, you will need more. International or private schools are especially expensive.
A rental flat for a family of four in Stuttgart can cost anywhere between €600 and €2000 per month (and you get what you pay for - so don't expect anything fancy at the lower end!). Houses cost more than flats.

In addition to what others have answered here I would add that the housing market in German cities is very tight and especially challenging for arriving foreigners. One usually needs help finding a place from the employer or to have temporary accommodation until one finds a place. Almost no Germans will rent to someone from overseas before they arrive and can meet them. Also, houses are not just more expensive than apartments but much harder to find. In and around cities, the vast majority of accommodation is simply apartments rather than stand-alone houses. And when Beppi says private schools are expensive this means around 12000 Euros a year per child compared to free for the usually good public schools. But the subjects of accommodation and schools has been covered extensively on other threads on this site.