Always remember that you are dealing with a very immature young animal. Be realistic, flexible, patient and always fair during puppy training sessions. Your puppy doesn't just automatically know this stuff! It's all new to him and he is bound to have the odd slip up and mistake along the way. Don't worry about these mistakes, just move on and do your best to prevent them in the future.
“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.
The foundation of training should be based on positive reinforcement. Positive reinforcement is the process of giving a dog (or person!) a reward to encourage the behavior you want, like getting a pay check for going to work. The idea is not to bribe the behavior but to train it using something your dog values. Avoid using punishment such as leash corrections or yelling. Punishment can cause a dog to become confused and unsure about what is being asked of him. It is important to remember that we can’t expect dogs to know what they don’t know – just like you wouldn’t expect a 2-year-old child to know how to tie his shoes. Patience will go a long way in helping your new puppy learn how to behave.