Like, if you massively accelerated the Moon's orbit or propelled it somehow so that it moved from one side of the Earth to the other in a very short period of time, what effect would that have on the people down on Earth? A lot? Any?
Probably not a lot - you'd probably a wierd high speed tidal cycle while it was moving, but once its on the other side, high and low tides would still be at the same time (apart from moon moving time, unless you keep going until it's opposite where it would have been) (there's a water bulge on both the side facing the moon and the side opposite, though the heights may differ. Not sure whether there'd be side effects - with the tidal pull shifting very rapidly, there might be a change of triggering some earthquakes/fault line slippages - though I'm not sure if the force would (a) be big enough to be a factor, or (b) whether the rate of shift would make any difference - if not, then just business as usual. Maybe some adverse results from dragging the tidal bulge around faster - could pump more water through restrictions, causing higher than normal tides or floods.
Moon phases would shift by half a cycle (new when previously full).
Similarly, solar eclipses would shift - old predictions would no longer be valid, and you'd have a new saros cycle. Depending on how carefully you move the moon, you might no longer get old style total eclipses - they're fairly closely determined by the earth moon (and sun) distance; too far out, and the moon no longer covers the sun completely, so you get annular eclipses; too far in and it blocks some of the corona.
Unless you rotated the moon as you moved it - then slowed the rotation rate back to normal at the end - we'd end up with a different side of the moon facing us. Fun for astronomers :). Since the moon is tidally locked, presumably it'd either drift back to the old position (but is likely to rotate or oscillate for a (Very?) long time until it eventually settles down again. So again, more fun for lunar observers.
Not sure whether it'd have a significant impact on satellites - I'd guess probably not, since it doesn't mess things up at the moment.
As well as a liquid tide (seawater moving about), the Earth has a solid tide - the rocks of the crust rising and falling. These solid tides seem to be what triggers aftershocks from earthquakes.
If the moon suddenly races to somewhere else, instead of taking its usual hours to get there, lots of rocks will be under abrupt stresses and strains. So perhaps it might trigger initial earthquakes, and not just the aftershocks.
The tides would be weird for a while, but not catastrophically weird.