If you fly from one 'Shengen' country to another (i.e. France, Italy, Germany... Sorry, I don't know the list), you will not have a single control.
However, if you go to UK, you are required to have a valid passport (an I.d. would be enough for a E.U. citizen, but as a U.S. citizen you will need a passport)
If you have a flight let's say JFK-CDG-TLV (so U.S., then E.U. then something else), well.... I don't know if you have to clear passport. I did though have to do it last winter when I flew CDG-MIA-SDQ, but I don't know what the policy is in Europe.
Same thing for the U.S.-E.U-E.U. flight, although I'd say you go through passport control at your arrival in E.U. as there will probably be no control after the second flight.
Now, if you cross French-Spanish border by bus, I can guaranty you that you will almost certainly get no passport control.
Hope this helps