There would be a big difference in moving a regulation, full size pool table, and one of the coin operated models.
If it's a 9 foot regulation sized table, the slate will indeed be in three pieces, and you'd want to disassemble the table carefully, and re-assemble properly.
If it's a coin operated table you can move it with the slate in it. The 7 foot coin operated tables (Valleys, Dynamos) have single piece slates). Just remove all of the balls, then lift up one side of the table and put a chair under that side, then get underneath it and screw the leg levelers (feet) out of the legs and set them aside.
Then take a flat dolly (like a piano dolly) and set it on the floor on the side where you've removed the feet. You'll need at least THREE PEOPLE with good backs...
You and another guy get on the side opposite the dolly and tip the table up on its side so that it sits on the dolly. The third person will be centering the dolly underneath the table's side as the pool table comes up. (If you try to raise the table up on its side without first removing the leveler feet, you'll break the feet and/or legs).
Nest you will remove the legs from the table to make it easier to move through doors.
I'd get a low trailer to haul it on, preferably an enclosed trailer. You can haul them on a pick-up truck, but you've got to be able to muscle the table up into the truck. Since it's top heavy, it would be easy to turn over as you were trying to get it up into the pick up.
Just go easy, and be careful, and realize that if you drop the table and break the slate you've ruined it...
Dan