Two of the most common commands taught to a dog are sit and down. I always add stand to these two. The reason for this is that 'stand' can be extremely useful when visiting a vet or checking your dog. Often though, people forget about this command since their dog walks up to them and are already standing, so they ask for a sit.
3 commands, 6 movements
As a human, we think of these commands as concepts. Sitting is sitting down with your butt, standing is standing up on your feet and lying down is lying down on something. But for a dog, because they are not so good at generalizing, these commands should be thought of as 6 different movements.
Stand to sit
Stand to down
Sit to stand
Sit to down
Down to stand
Down to sit
When you give a command, the dog links the movement to the command. So you need to generalize from the different starting points to make it clear to your dog.
How to teach these commands?
When teaching these commands, there are two possible starting points.
Classically, the starting point was a 'sit' position (doing so, you teach your dog sit to down, down to sit, sit to stand, stand to sit).
So starting from a sit position, you take a treat and let your dog sniff it. Then, you lower the treat to the floor and then move it away from your dog. He'll follow the treat. The first few tries, your dog might stand up to get the treat, just start over from the beginning. When he lays down, reward him immediatly.
From this position, you can lure your dog back up with a treat into the sit position.
Starting from the sit, you can also teach your dog to stand. Just let your dog sniff the treat and move it forward at the height of your dog's nose.
And once again you can lure your dog in a sit position by moving the treat little above the nose of your dog.
Now this second method I use pretty often with puppies. This time, the starting point is the stand position. (You'll be teaching stand to down, down to stand, stand to sit, sit to stand)
When your dog is standing, show him the treat and move your hand in between his front legs. His nose will follow making him bow and now he only needs to put down his rear end to be lying down.
Returning to a standing position can be done with just the opposite movement. Raising your hand to the level of your dogs head making him stand up.
From stand to sit, you just raise your hand above the head of your dog so his nose moves up and his butt moves down and there is your sit.
Then, one last time, move forward to get back into the stand position.