First thing that is recommended is to find a reputable veterinarian and schedule an appointment. If you purchased your puppy from a pet store, check to see if they partner with a veterinarian and if they offer a free initial exam. The veterinarian will perform a general exam, measured the puppy's weight and start discussing vaccinations.
Puppies need a series of vaccines until they are 16 weeks of age. The series usually starts out with a DHLPPC (Distemper, Hepatitis, Leptospirosis, Parvo, Parainfluenza and Corona) vaccine at nine weeks of age. Some puppies are allergic to the Leptospirosis and/or the Corona portion of the DHLPPC and some veterinarians decide to give the vaccine that does not have these components. At 12 weeks of age, you need to booster the DHLPPC and get the first of two Bordetella (kennel cough) vaccines. At 16 weeks of age, your puppy needs the final DHLPPC vaccine, the second Bordetella and a Rabies vaccine. Your puppy is now vaccinated for its first year of life.
Vaccines are not all your puppy needs in the first weeks. Your puppy also needs to have its stool checked for parasites and be given a dewormer. Since puppies are susceptible to many diseases early in life it is a good idea to keep your pup away from dog parks, pet stores and anywhere that may have other unfamiliar dogs. Be sure to discuss potty training with your veterinarian as they have very helpful advice.
With all of these health checks, vaccines and the other information you are going to be bombarded with, do not forget to give your puppy a healthy diet. Most major brands of dog food offer a puppy diet. It is important to feed your pup a diet designed exclusively for puppies. Generally a dog should be fed puppy food until it is a year of age.
These are just the basics of owning a puppy. There are many other facets such as behavior training, potty training and spaying or neutering. Discuss these things with your veterinarian.