At what speed is our solar system expanding, and if it is, why don't the positions of the sun and planets change, or grow farther apart? Why is the Big Dipper still where it is?
Our solar system isn't expanding, because it's bound by gravity. Even though space is expanding, the positions of objects in the solar system stay the same because gravity pulls them back.
This is true for all gravitationally bound objects, even galaxies and galaxy clusters.