This is pretty big topic.
Measuring star distances can be done in a variety of ways. The closer ones are done by the parallax method, the old basic geometry. However there are
many others (I have a list next to me that was done by a professional astronomer and even he puts "various other techniques" at the bottom. They can find variable stars, such as Cepheids, RR Lyrae stars and eclipsing binaries.
For globular clusters they plot all the stars out on a Hertzpsrung-Russell diagram and that gives them the clue (somehow, not sure about this one but I know it gives age of the cluster as well!) and if they know what type of star it is and it's mass they can use the inverse square law for light diffusion (ok someone else has to help me here!!)
Galaxies are done by redshift of their spectrum.
For solar system objects you need to get a minimum of 3 positions and then some maths is done to compute the orbit. I'm still to learn how that all works as well, but the more observations the better the result.
Size of asteroids is fuzzy, as they can only go on how much light it reflects - ideally a large asteroid would reflect lots of light, but what if its got a dark surface? If the asteroid has a moon then you're laughing because the interaction of the two bodies gives the mass away.
Hopefully this will get the ball rolling and get some clearer answers coming in.
