My wife and I are thinking about living in Europe for 6 months, spending most of our time in France. I think you need a visa to live in France longer that 3 months? If we stay in France for 3 months, they move to Germany or Italy for a week or two, can we come back to France without a visa?
No! France, Germany, and Italy are all parts of the Schengen zone. You cannot stay in the Schengen zone for more than 90 days in any 180 day period. Leaving for a few weeks does not "reset" the clock. Even if you left the Schengen zone for a few weeks, you would still have been in the zone for 90 days out of the last 180. After being in the zone for 90 days, you would have to stay out of the zone for 90 days. The UK is not a Schengen country. You could plan on spending half your time there and half in France. Or, get a visa.
To stay in France longer then 90 days in a 180 day period, you would need to apply for an extended tourist visa from the French government. I don't know the specific requirements for France, but for these things you usually have to prove that you have sufficient income and health coverage to last the for the duration of the stay. Because there are no border checks between France, Germany and Italy, you can come and go as you please.
Those two answers above are right on. We assume you are US or Canadian Citizens?