I mean across the sky.
I think I could figure this out if I had some time to reason through it; but I never seem to be able to find the time when I’m thinking of the problem
So when watching the moon and the sun move across the sky, does the moon “catch up” to the sun once a month (i.e. the moon moves faster); or does the sun “catch up” to the moon (the sun moves faster).
Could you provide a basis for your reasoning too?
The Moon moves slower across the sky.
If you’re looking down at the Solar System from above, everything turns counter-clockwise, in orbit and in rotation (well, except for Venus and Neptune, but neither of those is relevant for this question). Because the Earth is turning counterclockwise, everything else moves clockwise relative to the Earth. Nothing else of interest has a period as short as a day, so this is the dominant motion in the sky.
But to say which things move faster, we also have to consider the orbital motions. The motion of the Earth around the Sun is fairly slow, once per year, so compared to faster things like the Moon, we can ignore it. But the Moon makes one orbit a month. And this orbit is counterclockwise, making opposite of the clockwise-relative-to-the-Earth motion that everything else shows. Since a one month period is slower than one day, it can’t completely counteract the effect from the Earth’s rotation, but it partially does so, so the net motion across the sky is slower.
Never mind.
The Moon crosses the sky very slightly faster than the Sun, because it orbits the Earth in prograde, that is, it’s orbit is in the same direction as the Earth’s rotation. We can show this in terms of degrees per second. Ignoring factors like the Earth’s orbital speed, the Sun makes a complete transit around the Earth in ~24.25 hours, or 360 degrees in 87,300 seconds (.0041 degrees/sec). To this, we add (because it’s in the same direction) the Moon’s orbital speed, which is one orbit in ~29 days or 360 degrees in 2,531,700 seconds (.00000039 degrees/sec). Not exactly jaw-dropping, but there you have it.
Never mind. This page explains things quite well.
What I’ve never been able to get my head around is, if the sun is moving at some fantastic velocity in a orbit around some point in our galaxy, are we on Earth not also moving at said fantastic velocity, in order to stay orbiting it? 
Yes, but it’s not that different, conceptually, from you in a train traveling at 100 MPH with a fly circling your head at 2 MPH.
Yes, but since everything on, and including, the planet we’re standing on is doing it, we don’t notice.
And even though it’s a phenomenal speed, the stars are so distant and far apart that we don’t notice ourselves whizzing by them.
On average, the moon rises 50 minutes later each day. Although there is drift with the seasons, the sun rises within minutes of the same time on consecutive days.
So it’s not like a fly circling a fully extended yo-yo outside a train window, while travelling at 100mph?
No, not at all. In that setup there’s a heavy headwind for the fly to contend with. But there’s no air, and hence no “wind”, in space.
The sun is orbiting the centre of the galaxy, and the earth accompanies the sun. It isn’t left behind since it, like the sun, was created from a cloud of dust and gas which was orbiting the galactic centre to start with - in this sense it could be said that both the sun and the earth are orbiting the galactic centre together, even though we don’t normally express it that way.
What if the train track was inside a vacuum, and the fly had a space suit on? 
Flies can’t fly in a vacuum.
I suppose you could fit a miniature rocket engine to the fly’s space suit, though.
How do you define “above the solar system” as opposed to below. What is your reference ?
Above the North Pole, duh. Why do you think we call Australia “Down Under?” 
I’ll be sitting over here waiting patiently to make notes on how you square that with the moon rising *later *each day. :dubious:
Neither the moon nor the sun move at all, except relative to other objects. Either one can become your “zero point” in any motive system. So the answer is somewhere between a big fat 0 and lightspeed.
As far as how fast they appear to move across the arc of the sky, well, that’s something else entirely.
See post #5.
Gah, did I actually write that? I meant “north”, not “above”.
I deleted my earlier post because it was too confusing. It is true that the Moon moves fastest against the background sky (GHA) but it is also true that because of the rotation of the Earth the algebraic addition of both movements results in slower local motion (LHA) than that of the Sun or stars.
I find no contradiction in thinking [moon = fast moving] and [moon = slow moving] in diferent contexts but I understand how this can be confusing to the layman.