“Down” can be taught very similarly to “sit.” You can wait for your dog to lie down (beginning in a boring, small room such as a bathroom can help) and capture the behavior by reinforcing your dog with a treat when he lies down, giving him his release cue to stand back up (and encouragement with a lure if needed) and then waiting for him to lie down again. When he is quickly lying down after standing up, you can begin saying “down” right before he does so.
Once your dog can stay in a sit for several seconds, you can begin adding distance. Place him in a sit and say “stay,” take one step back, then step back to the pup, give a treat, and your release word. Continue building in steps, keeping it easy enough that your dog can stay successful. Practice both facing him and walking away with your back turned (which is more realistic).
Once your dog can stay, you can gradually increase the distance. This is also true for the “sit.” The more solidly he learns it, the longer he can remain sitting. The key is to not expect too much, too soon. Training goals are achieved in increments, so you may need to slow down and focus on one thing at a time. To make sure the training “sticks,” sessions should be short and successful.
The first method is called capturing. Stand in front of your puppy holding some of his dog food or treats. Wait for him to sit – say “yes” and give him a treat. Then step backwards or sideways to encourage him to stand and wait for him to sit. Give another treat as soon as they sit. After a few repetitions, you can begin saying “sit” right as he begins to sit.
Imagine your puppy running out the front door. You call him, waving a treat. But he'd rather chase a squirrel into the road than come back to munch on a treat. In addition to the obvious danger of Puppy getting hit by a car, he learns that he doesn't have to listen to you. He learns that he's in charge of what he decides to do and what he decides not to do.