Whether you have an adult cat, or a new kitten, teaching him how to use a cat door will be done the same way. When you install your cat door, make sure you measure the "rise" of your cat (the measurement from the floor to the lowest part of your cat's chest or stomach). This measurement tells you where to place the "bottom" of your cat door. The bottom of your cat door should be an inch or two lower than the "rise" of your cat.
If you have a puppy you will need to install the cat door flush with the ground; and you will need to re-install it at higher intervals as your puppy grows. Another option is to take an educated guess as to how tall your cat will eventually be, install the cat door at the appropriate height, and construct a "kitten-ramp" so your kitten can reach the cat door and go through it comfortably.
Once the frame of the cat door is installed in a wall or door, leave the "flap" off at first. Have someone stay inside with your cat while you go outside. Call your cat through the "hole" (cat door frame without the flap). When he goes through and comes to you, praise him lavishly and give him a food treat. Now have the person inside the house call him through the "hole." When he gets to them, they should praise lavishly and offer a food treat as well. Do this at least 3 times and no more than a dozen. After this, your cat will know there is a hole in the wall or the door especially for him.
Leave the "flap" off the pet door for one full day. Encourage him to use his cat door by not letting him use the "real" doors. Instead, you use the real door and say to your cat, "Go to your door!" pointing in the direction of his cat door. You may need the help of someone inside to "help" the cat find his new door. After a half a dozen times, your cat should like this new game! If you have a very young puppy, do not expect them to learn "Go to your door" for many weeks or months; still give them the command in a happy voice, and have someone inside show them where their door is every time. It sometimes helps if you are outside (after going through a real door) and someone else helps your cat or puppy find the cat door as you call him from outside.
On the second day, install the "flap." Now, you will need to repeat the same exercise as when you first sent your cat through the "hole." But this time, the person on the same side of the door as the cat will need to "push" the flap open for him. Each time the cat goes through the door, push the flap less and less for him. It is important that the cat gets used to the feel of the flap on the back of his head so once your cat has begun going through the door, let go of the flap so he feels it on his head and body as he goes through the door. Eventually the cat will need to push the flap by himself and cats are usually hesitant to do this at first. He will probably put his nose down by the bottom of the flap and wait for the flap to move (after all, it has up to now).
At this point, push the flap slightly so that your cat can see it is a moveable object, let the flap bounce back to the closed position. The best way I can explain it is that you are "poking" the flap using short, quick pokes. This gives the cat a glimpse of an opening and encourages him to poke the door himself. At this stage, some cats begin going through the door with ease, others become quite excited, but still haven't figured out that they can push the door open.
If your cat will not push the door open by himself yet, secure the bottom corner of the flap to the flap itself, or above the cat door using tape, string, or anything else that works. You want the flap to be on the cat door, but the corner turned up so that the cat can see a small opening. He should then feel more comfortable pushing the flap open on his own. If your cat needs the corner of the flap turned up, leave it turned up for 1-3 days until your cat is very used to using his cat door. After 1-3 days, do the exercise again with the entire flap in place.
After your enthusiastic encouragement and praise, your cat should be able to push the flap now with no problem.