Shouldn’t the moon have… 24 time zones as well, depending where on the moon you currently are?

No, the moon’s rotation isn’t on a 24-hour cycle. I’m not a astronomer, but I pretty sure since it’s tidally locked to earth and on a 28-day cycle around the earth, a lunar day is actually 28 Earth days, but I’m not actually sure how that would factor into the number of time zones (but I’m pretty sure it would be more complicated than just 24 time zones to match 24 time zones on earth).

Plus, I think the speed of the moon relative to the sun is different enough from Earth that you need to take relativity into effect, which is the real headache here.

Yep, relativity accounts for a difference of like 50ms drift per earth day. I would assume that it’s forward drifting if you’re on earth but backwards if you’re on the moon.

Take that, timezone whiners!

It’s been a long time since I took modern physics, so I’m not positive, but I think you’re right that the moon would have time moving slower, and if your 50ms/day is right, then you’d need to do something like skip a second every 20th day on the moon to keep pace with Earth. We could call it an “anti-leap-second”