New puppies should get their first set of shots at 6weeks of age, not before.
Then every 3-4 weeks the puppy should get booster shots, not less than three weeks as it may lead to immune system reactions.
The immune system matures and maternal antibodies fall off around 12 weeks of age, so the puppy must have a series of two vaccines 3-4 weeks apart at or after 12 weeks of age. For most puppies I recommend vaccines at 6, 9, 12, and 16 weeks of age. If some are missed, we don't go back and make them up, we just hope the puppy didn't get one of those diseases in the meantime when he/she was unprotected due to a missed vaccine. If a puppy goes unvaccinated until 12 months of age or longer -- it still needs a series of two vaccines 1 month apart to bring up the immunity to those diseases. (ie still needs the 12 and 16 week vaccines no matter how old the puppy is.)
Distemper combination vaccine is a "4 way" vaccine and it has been for decades. It protects against distemper, hepatitis, parainfluenza, and parvovirus. This is typically given at 6, 9, 12, and 16 weeks of age. Leptosporosis vaccine is often added in at 12 and 16 weeks of age depending on the risk in the place the puppy is going to live. When a rabies vaccine can be given varies on the laws of the region, but 16 weeks of age is ideal, but legally must be administered by a licensed veterinarian.
The earliest a dog or cat should be vaccinated is 5-6 weeks of age. Puppies shouldn't be separated from their mother until ideally 8 weeks of age, to allow for appropriate puppy socialization with the mother. I believe it's the breeder's responsibility to make sure 1) the mother is well vaccinated prior to pregnancy before breeding, 2) that the first series of vaccines be provided by the breeder (at 5-6 weeks of age). The remaining vaccines every 3-4 weeks should be then followed up starting around 8-9 weeks of age, depending on when the first vaccine was given. Any vaccine PRIOR to 5 weeks of age doesn't typically work, due to interference with the mother's natural antibodies that are in the puppies. The rabies vaccine shouldn't be given this early and may not be legal, depending on what your state requirements are.