“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.
Just as a child needs a caring parent; an athletic team needs a coach; your puppy needs a leader and a clear social hierarchy. If you do not take up the role of leader, your dog will; and you will end up with an unruly, disobedient dog. Many people try to win their new puppy's love by letting the puppy always have its way. Buckets of affection is a wonderful thing for most puppies, but it must be tempered with respect.