“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.
First, make sure your puppy is comfortable wearing a leash. This can feel strange at first, and some puppies may bite the leash. Give your puppy treats as you put the leash on each time. Then, stand next to your puppy with the leash in a loose loop and give him several treats in a row for standing or sitting next to your leg. Take one step forward and encourage him to follow by giving another treat as he catches up.
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.