What do we call a scientist who studies the stars?

An astronomer. The distinction from astrology is recent - the two were the same job for most of history, and Kepler cast horoscopes for a living while working out that the planets move in ellipses. He described it as the foolish daughter supporting the wise mother.

It is the oldest science because the sky was the only reliable clock and calendar available. When to plant, when the river would flood, when the rains were due - all of it came from watching, which is why so many prehistoric monuments are alignment instruments and why almost every early state paid people to keep sky records.

For most of that history the work was done with the naked eye and done superbly. Tycho Brahe measured planetary positions to about an arcminute with no telescope, and it was precisely the accuracy of those measurements that let Kepler prove the orbits were not circles - a smaller data set would have fitted a circle fine. The modern discipline mostly studies things the eye cannot see at all: radio, infrared, X-rays, gravitational waves, and a great deal of matter that emits nothing whatever.