TL;DR
Core vaccines (DHPP and rabies) are non-negotiable. Bordetella, leptospirosis, and Lyme depend on where you live and how your dog lives. Puppies get boosters every 3-4 weeks from 6-8 weeks old until 16 weeks.
The core vaccines every puppy needs
DHPP protects against distemper, hepatitis, parvovirus, and parainfluenza. Parvo alone kills about 90% of untreated puppies — this is the vaccine you never skip.
Rabies is legally required in almost every US state and is given once around 16 weeks, then boostered.
The lifestyle vaccines — it depends
Bordetella (kennel cough) is required by most boarding facilities and doggy daycares. If your puppy will go to either, get it.
Leptospirosis is transmitted by wildlife urine — even in cities with rat populations. Ask your vet what's endemic in your area.
Lyme disease matters if you're in the Northeast, upper Midwest, or hike in tick country.
The rough schedule
Most puppies see the vet at 8, 12, and 16 weeks. DHPP is given at each visit; rabies at the last. Lifestyle vaccines get added into that schedule based on where you live.
