“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, 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.
Teach a cue. Introduce your puppy to a sound cue that means, “food is coming.” Some people like to click and treat, some people use a word like “yes,” and some people cluck their tongue. Whichever you use, the method is the same: In a quiet, distraction-free area, with the puppy on a leash and collar, make the sound. The second your puppy turns toward you and/or looks at you, reward him with a treat. After a few repetitions, you’ll notice your puppy not only looking at you, but also coming over to you for the treat.