Let's say you have fifty different units on the board. When it's your turn, you get to move each of them. How do you keep track of which units you've moved already and which ones you haven't?
Avoid unnecesary moves in the combat move phase
Belive it or not, I found out that the typical source of confusion is that most players move units that are not going to fight (nor Aircraft carriers that won't provide a landing base to attacking airplanes) in the combat move phase.
It's not against the rules but.. there is no benefit in doing so. In fact it's much better to wait for the result of the combats to know where to move.
Never doing that helps a lot: every unit that is not in a combat area, has not move prior to non-combat phase move.
The airplane combat nightmare
This happens with japan and germany, specially in 1940 editions. Even though you know that airplanes have moved to the combat zone you need to keep track of the ammount of movement left
There is an idea introduced I think in the revised edition: fuel markers. Simply, have a die or any other thing that can mark from 1 to 5. When you gather many airplanes in one zone start adding one marker for each airplane, each showing the ammount of movement (fuel) left. Separating of course each kind of airplane.
When selecting airplanes casualties simply remove the lowest fuel marker (that's called being a good captain) so you are sacrificing airplanes with the least movement.
On the non combat move phase move the remaining airplanes and each fuel marker shows the movement of any given airplane.
Have your enemies pay attention
This is a gentlemen's game. They can tell you what has moved and what not. They know because they are looking at your positions for possible attacks all the time
Hope any of this helps!
Any marker token that can be distinguished from the marker tokens (gray/red) already included in the box will do. This could be something as simple as pennies, or other small change, glass beads, wooden cubes, etc.
I had a similar problem with a chariot racing game, so we used coins to mark the hexes that the chariots moved from, just in-case a player changed his mind about a move and wanted to know where their piece started.
I use a pattern where I move the units on the front(s) first and work my way backward toward my industry(ies). I also picture a front like a line of territories followed by other lines of territories. When moving I can then take care of each 'lines' one at a time.
This way I never move units into a territory where local units will have to be moved later and, most of the times, I only have to manage each territory once. This prevents the confusion of the back-and-forth.
Also to avoid confusion, I never leave a chip(s) without a unit piece on top of it. It's more manipulations but it's much safer.