How does the sun stay in one place?

Answer

The sun and the planets (with their satellites) revolve together around their common center of mass. The greater the mass of an object in a system with many objects, the closer it is to that center of mass. The mass of the sun is about 300 thousand times that of the earth, so that the sun is only a 300 thousandth farther from the common center of mass. That is then about 500 km, while the radius of the sun is 700 thousand km: so we can rightly say that the earth revolves around the sun. But you’re right: don’t let the sun stand still completely.

In our solar system we describe the movements relative to the center of mass of the whole system. In such a reference system, the center of mass is therefore a fixed point. But that does not exclude the possibility that that center of mass, and by extension our entire system, in itself also performs a movement with respect to something else. And indeed we find that our solar system describes a great circle around the center of our galaxy. The radius of the orbit is about 25 thousand light years, the orbital period about 250 million years.