By using radio transmissions time, and radar reflections off another planet, images of stars in telescopes, astronomers can fix position of Earth around the sun to within a few feet. This is necessary to calculate and control the paths of probes to moon, asteroids, and other planets.
As far as going around the Milky Way, the solar system, an entity almost a lightyear wide, takes 220 million years for one trip. Astronomers can only fix the relative position of the sun to other stars (also moving) to within just a fraction of a light year, not much better. A projection of the path is uncertain because of inaccuracies of measurements and the great distances involved.
The Milky Way galaxy is in a group of many. Its companions, the LMC and SMC, and some other fainter dwarfs are proceeding toward M31 galaxy in Andromeda, another large spiral. The glancing collision that will happen in a few billion years was recently updated with a better estimate.
The Local Group of galaxies is moving toward the Virgo cluster of even more members. That bunch is moving toward "The Great Attractor" somewhere beyond Virgo. Exact measurements on position and directions are not possible now. Speed estimates can be made from Doppler effect on light beams from stars.
65 years of studying Astronomy. My cousin's husband works at JPL. He was on communications teams for planetary probes. Now he is a manager for the JPL giant radio telescopes around the Earth.