I think harnesses are really great for small dogs especially. It's good to expose them to harnesses at that age, and it takes pressure off the neck. Especially if the dog is small, it should be wearing a harness unless it doesn't pull. Small dogs are at greater risk for collapsed trachea.
I would start by doing leash pressure games, not even walking. Most puppies and animals have no idea what a leash is. When the leash is tight, they feel trapped and they try to escape. They are not 'fighting you', it is just a natural response. I teach pups to come IN the direction of pressure to release it and get a food reward. I start with so little pressure that the puppy is not fighting it at all. If a puppy is planting or straining away, it is too much too soon.
Once the pup knows to approach me to 'get rid of the pressure', the leash walking comes naturally.
If your pup is not fighting the leash and is just lying down, do not pull but also do not let him back up. This is assuming your pup is good at the exercise above. Then, just turn your back to the pup and wait. The moment he comes along, reward him. Too often, owners 'teach' puppies to plant themselves because puppies learn that planting makes owners bring out treats to coax them.
