How can you measure the motion and mass of the sun?

Answer 1

Measuring the mass of the Sun is easy if you know Newton's gravitational constant, G. The mass of the Sun is then determined by the velocities of the planets in their orbits.

The motion of the Sun---well, with respect to what? The motion of the Sun relative to the Galaxy is determined by measurements of all the other stuff in the Galaxy. One of the more accurate things to measure is 21 cm atomic hydrogen line, in all directions. The velocity of thousands of nearby stars is also kwown. One of the more accurately-known velocities nowdays is the velocity with respect to the Cosmic Microwave Background. This is determined by fitting a dipole term to the measurements of the WMAP satellite.