Puppies are generally weaned after 4 to 6 weeks. It is not uncommon for puppies to be available right after they are weaned. Your new puppy should be at least 8 weeks old. Puppies learn body language and dog social skills from their mother and litter mates from 4 to 8 weeks of age. This is very important from a training perspective. Bite inhibition is one of those social skills. The puppies mother teaches him not to bite. You would rather he learn from mom than have him draw blood the first time he mouths your hand. They learn how to play. And that includes what is a play bow versus what is a dominant stance. This body language knowledge helps the puppy when meeting other dogs.
Remember your new puppy does not come home house trained. So you have to have a place that can tolerate accidents. (Carpeted areas are bad). Every puppy likes to mouth and chew whatever he can get his teeth on. You need to make sure whatever the puppy can reach, you would like teeth marks in. Consider a baby gate to keep the puppy out of parts of the house. Puppies will chew on wood work, area rugs, wires and extension cords. You want to have toys that you want the puppy to chew on.
Cleaning supplies to clean up accidents. Do not use ammonia based products. Urine has ammonia in it!
Multiple Toys - at a minimum, a fuzzy toy and a chew bone. You don't know what your new puppy will like to chew on.
Crate - this can be used as a safe zone for the dog. (This requires you to teach your children to leave the puppy alone when he is in the crate). Buy a crate that will fit your full grown dog in, then use a box to make the inside smaller to fit your puppy. There should be enough room for the puppy to lay down, sit, and turn around. Too much room it is possible the puppy will use a corner of the crate as a bathroom and this will make house training more difficult.
Water dish and food dish - if you have more than one dog make sure they each have a food bowl
Lead and Collar that fits - expect to buy multiple collars. Do not buy a collar that he will grow into.