Julian, quite a challenging question. I will only be trying to answer the first part – the Sun-Earth distance and the Earth-Stars distance.? Even then as my team suggests, I might be introducing mathematical terms that you have not met yet, but I have included links to other sources of help.
To answer the first question I recommend you read a Universe Today article It is an excellent historical review of the problems that the early scientists had in determining the Earth-Sun distance. The answer finally came from observations of the movement of the planet Venus across the face of the Sun. In it the writer refers to a Nasa document that tries to explain the methods used. In present times the distance to the Sun is measured by ‘bouncing’ a radar pulse of of it.
Determining the distance to the other stars becomes possible once the Earth-Sun distance was known. It uses a technique called parallax. I would like to illustrate this with a question which tackles a simpler problem. ‘How far is my finger away from my nose?’
Try this little experiment, put a finger in an upright position in front of your nose. Now close one eye and note the position of the finger. Close that eye and open the other one. The finger moves! Now suppose, with help, you could measure the amount of movement. You could end up with diagrams like those below. Did you make a note of the position of your finger relative to your nose? No – you can now see how you could work this out.
Now let’s do a little geometry and add an axis
We can then measure the angle of the apparent movement
You end with a right angled triangle ABC, knowing the angle x AND the distance between your eyes you should be able to do a bit of trigonometry using TAN x = opposite/adjacent (Tan x = AB/BC) and work out the distance of your finger from your face. For an introduction to trigonometry please look at this site.
Amazingly this is (in a crude way) the same process by which astronomers can measure the distance to the stars. Instead of using the distance between your eyes they use the orbit of the Earth. They look at a star and make a note of it’s position and then do the same thing 6 months later when the Earth is at the opposite side of the Sun. They therefore have AB (the distance between the Sun and the Earth and they have the angle through which the star has apparently moved.
This gives the route to determining the distance between the Earth and a Star.