If I were to look at the Moon through a powerful telescope, could I see the Apollo landing site?
Not really. The Moon is huge. The Moon is a little more than a quarter the size of the Earth, about 2100 miles across. In fact the Earth-Moon system should really be called a double planet. The largest telescopes resolve things roughly 1/10 of a mile across at the Moon's average distance. That means they can't really see things smaller than blobs about 500 ft across, and doing even that well requires some serious telescope horsepower.
What is the biggest planet?
In the system of planets circling our sun, the biggest planet is Jupiter. It is about 11 Earth diameters across at its equator, somewhat less through its poles (Jupiter is flattened by its high rate of rotation).
How big is the Sun?
The sun has a diameter of about 100 times that of the Earth, or approximately 800,000 miles across.
How big can planets get, even in other star systems?
While we have never seen a planet in another star system really well, we have models of what a big planet mostly made of gas will do when you add more gas. It turns out that Jupiter is about as big as it is possible to become. As more gas is added, it just sinks in and makes the planet more dense. Somewhere between 50 and 100 Jupiter masses, the center of the planet becomes compressed enough that the planet makes the transition to a dim star.
How big is our galaxy?
The distance from the Sun to the center of our galaxy is as far as light would go in about thirty thousand years. We are on the outskirts of the galaxy. The whole galaxy is thus somewhere around 100,000 light-years across. It is much thinner though.
How big is the Universe?
From any location within the Universe, the Universe appears to stretch out to a radius of somewhat more than 12 billion light-years.
I have heard that the Universe has no center. But isn't the center of the Universe just the average position of the galaxies? What is beyond the edge of the galaxies but more empty space, forever and ever?
The problem with doing such "averaging" is that as we look out across the galaxies, farther and farther, we are looking back into time. Light is so slow that it just cannot bring information to us fast enough. We see the edge of Universe as a limit in time rather than space. If you were to jump into a fast spaceship and travel in a straight line, you would never reach a place where you can look out over the edge of the void. So, in a way, the Universe is finite and infinite at the same time. I find the fact that it is impossible to fly to the edge of the Universe strangely satisfying.
How are such distances as light-years measured? After all, you can't go there with a tape measure.
Nearby distances (out to a couple of hundred light-years) are measured by means of triangulation, or the amount the stars shift against the background with the motion of the Earth during a year. It's kind of like binocular vision. There are enough stars out to that distance that we can calibrate other distance scales, like that of Cepheid variable stars. Cepheids vary their light output over periods of days to weeks. These stars have a luminosity that is calculable from the periods of variation. Thus, if we see a Cepheid too far for triangulation, we have only to measure the period and we know how bright it is. Then we measure the amount the light has dimmed over distance and we then know how far it is. Even Type 1 supernovas have been used to set up a calibrated distance scale.