Does the Moon revolve around the Earth, or do they spin around each other as they revolve around the sun?
The Earth and the Moon both orbit around their centre of mass.
The center of mass of the Earth and Moon is called the Earth-Moon Barycenter or EMB. It is about 1,700km beneath the Earth's surface. Both the Earth and the Moon orbit around the barycenter.
The Earth-Moon barycenter orbits the solar system barycenter.