And why isn’t the Earth moving at the same speed?
The most intelligible response here seems to suggest it is 250km/s, or 0.1% of the speed of light.
How is it then that the Earth is only orbiting at 32,000mph( or so) if it is following the sun’s orbital path? Does speed have a different meaning when great distances are involved? I really have trouble getting my head round this, so I hope you’ll go easy on me.
What are the basics I need to know to help me relate to what is happening here?
The two numbers are not measuring the same thing, one is the sun around the galactic center, the other is the earth around the Sun.
It’s all relative. You have to measure speed relative to something. So the Earth orbits the sun at X. The sun orbits the galactic center at Y. The speed of the Earth relative to the galactic center would be a lot closer to Y than X, varying depending on whether the Earth is heading towards or away or sideways relative to the galactic center at any given time.
So if you measure both against the galactic center, both are moving way faster than 32000 mph.
The 32,000 mph measurement only measures the earths speed in relation to the sun.
A quick mental exercise. You are on a train that is traveling 100mph. You walk from the back to the front at 5mph, and then return to the back at the same speed.
Your walking speed is 5mph. Your total speed is 105 mph while walking forward, and 95mph while walking backwards.
If you just want to know how long its going to take you to get from one end of the train to the other, then figuring out your total speed is useless… All that is relevant is how fast you walk. That doesn’t make any of the answers incorrect though.
So anyway, the earth is moving at 32,000 mph around the sun. The sun is moving at 560,000 mph around the galactic core. This means that at one point in earths orbit, the planet is moving at 528,000 mph in relation to the galactic core, and at the opposite point of its orbit, it is moving 592,000 mph.
But it still travels around the sun at 32,000 mph, just like you move 5mph when walking on the train.
So the Earth rotates on its axis. And revolves around the Sun. Wile following the Sun around the Galaxy. And the Galaxy is flying away from everything in the Universe.
No wonder I feel dizzy.
So it’s reckoned.
If you consider those two motions together, you’ll realize that the earth doesn’t rotate around the sun relative to the center of the galaxy. It’s getting dragged around that center with the sun, but it’s kind of just wiggling to and fro with respect to the center of the galaxy. So its motion in the same (general) direction as the sun is approximately the same speed as the motion of the sun around the center of the galaxy.
If you really want to get dizzy, consider how fast our galaxy is flying away from galaxies 15 Billion light years away…or at least how rapidly the space in between our galaxies is expanding over that distance.
Um, please note, all of you, that since I am the center of the universe, it all revolves/rotates around me. 
God I love Einstein and Relativity!
And yet for some reason, everyone’s moving away from you . . . 
About twice that. I’m guessing you did pir instead of 2pir.
Well, I’m good for the simple math. I only really understood Relativity for about twenty minutes once back in the '80s. I remember there was a really useful graph.
The Lorentz transformation? Og, I hated that thing in intro physics.
Michelson and Morley felt the same way 100 years ago. Unfortunately, they only made things worse!
Let’s be clear here …
Relativity, Einstein, Lorentz, Michelson & Morley, et al have exactly zero to do with the OP’s confusion & introducing them will simply make him even more confused.
Try this, Mr. OP …
When you sit in a chair you aren’t moving (e.g. your speed is zero) when measured against the room you’re in. When that room is really a railway car running down the track, your speed is still zero compared to the car, but it’s now also 60mph when measured against the countryside outside the window.
Any speed number anyone ever gives you anywhere anytime is meaningless unless they also tell you … measured against X. (relativists, hush up, I’m simplifying)
And that means that at any moment you can be going at different speeds when measured against different things.
In day to day conversational use, we assume speeds are measured relative to the surface of the Earth. But that’s just a (bad) habit of English, not a fact of Physics. And it accustoms us to forgetting the other half of a speed is always “measured against something”
I’m gonna make up some numbers here for the sake of example; I don’t have the time to go Google up the actual factoids.
So while you’re sitting in your chair moving zero mph measured against the railway car, you’re moving at 60 mph measured against the countryside, ~800 mph measured against the center of the Earth, ~32000 mph measured against the center of the sun, 70,000,000 mph measured against the center of the galaxy, and and 13,000,000,000 mph measured against some other galaxy far, far away.
All at the same time. To say “Your speed is 50mph” without mentioning “measured against what” is like saying “the final score was Manchester United 2” without naming the opponent or their score.
[nitpick]
The sun and the earth actually both orbit around a common center of gravity. Given their relative sizes this shared center is darn close to the center of the sun but not quite. Indeed it is this fact that has allowed us to discern other planet’s existence in distant solar systems. The distant start appears to wobble a bit as it orbits a common center of gravity with another object in its solar system and that wobble can be detected.
Of course since we have multiple planets in our solar system all tugging on the sun makes things a bit more complicated but the above is the general idea.
[/nitpick]
I think the Michelson/Morley experiment has everything to do with the OP confusion. They attempted to prove the existence of an absolute point of reference for the universe, the so-called Luminiferous aether which, had it existed, would have allowed them to measure the absolute velocity of the Earth, which is what the OP was asking about.