As a US citizen, you are permitted to stay in the Schengen Zone without a visa for 90 days in any 180 day period. Leaving Schengen and returning will not reset this.

Some countries in the EU are not in the Schengen Zone. Some countries in the Schengen Zone are not in the EU.

If you need a Schengen visa, you should apply to the Schengen country where you will spend most of your time, regardless of where you plan to enter and exit the Schengen Zone. If there is no country where you will spend the bulk of your trip, then you should apply to the country of first entry. Before doing this you should research the requirements for being a long-term visitor.