Sulfureous's answer is very similar to the way I've taught cats to use a cat door.
To some extent it depends on the cat: some of the cats I've lived with figure out with little more than me pushing the flap that they can go through it; others need the stage of having the flaps taped up (or removed) first.
Once the flaps are down, I put something they really like and can smell on my side of the door and the cat on the other side, then call the cat through. Some cats need extra hints here and can need the extra reinforcement of me pushing the flap to show them that it can move.
If I'm trying to move their litter to the other side of the door, I'll also move their food there - preferably in a location where they have to go past the litter box to get to the food.
The biggest problem I've had so far is with a very timid cat who's afraid of the basement. That one hasn't been resolved yet because we're keeping a litter box upstairs for the 18+ year old cat whose arthritis makes her reluctant to use stairs at all (We're also feeding upstairs because of this). When she passes, we'll be needing to get the fraidy-cat comfortable with the basement so we can move all the kitty litter and food downstairs.