cosmology - At what rate are stars accelerating?

As has already been mentioned, stars within a galaxy do not generally expand. The stars in a galaxy are gravitationally bound together.

But as you observe more and more distant galaxies, you see a general trend that distant galaxies are moving away from us at a speed that's proportional to their distance. (There are some small variations due to the random motion of individual galaxies; for example, the Andromeda galaxy is approaching us, and will collide with the Milky Way in about 4 billion years.)

This does not indicate that everything is expanding from some central point. An observer in a galaxy a billion light-years away would see the same thing, all other galaxies receding at a speed proportional to the distance from the observer. This can be interpreted as space itself expanding. Going into more detail would require General Relativity, which I'm not competent to explain.

Your question is about the rate of expansion. The answer to that is what's called the Hubble Constant, which describes the ratio between a galaxy's speed of recession and its distance from us. The current best estimate of this ratio is 67 kilometers per second per megaparsec. A parsec is about 3.26 light-years; a galaxy one million parsecs away will be moving away from us at about 67 kilometers per second.

Source: https://en.wikipedia.org/wiki/Hubble%27s_law#Determining_the_Hubble_constant