A start on puppy and kitten basics.

Vaccines - every 3-4 weeks from the age of 2 months to 4 months, we use this to see your new pet and look for issues before they develop.  They require at least 2 boosters, so that the vaccine actually results in protection from the disease being vaccinated against.  The first rabies is given on or after the 4 month visit.  These important vaccines will them be boostered again a year later and as recommended after that (usually yearly for DALPP and bordetella and 3 year for current rabies nowadays).  No animal is ever "done" with vaccines except in extremely rare cases.
Deworming - every 2 weeks for 2 doses in low count areas (CO and other dry states), these liquid dewormers are often pre-empted into Heartworm preventives each 4 weeks.  Make sure something is being given, as these parasites can cross species and cause some serious illness.
Heartworm - "more important" to canines (puppies and adults), this vector-borne disease (from musquitoes) can infect cats as well.  The preventives are usually pretty easy to give and very effective.  Ask your vet about them, as some breeds and areas have different preferences and needs.  After 6 months of age all pets are tested to prove a negative status before starting a preventive.  The most recent guidelines recommend year-round administration even in seasonal areas to decrease parasite resistance and spreading of the disease by sub-clinical carriers.
Socialization - This is in it's simplest form touching and introduction of people, pets, and others that the pet will not always be around but needs to get used to.  Obviously, some of this is within the family, but much is outside of it.  Puppy classes are a great way to start and interact with many animals and people.  An unsocialized pet is often fearful, noisy, and aggressive to strangers.  Working through certain pet issues will often require a trainer or behaviorist so that the inappropriate behaviors aren't reinforced.  Catching it early and correctly is the best strategy.  Any behavior that you wouldn't want in the adult, should be trained out from teh pup.  Positive reinforcement is the best in all species, since the young want to please and get good feedback from interactions.  Everyone in the family and around must be on the same wavelength when training.
Grooming - start early, start slow, and repeat often.  Under 3 months do only sections of less than 1/3 of the pet.  Whether brushing, trimming or bathing this keeps them from being overdone and stressed.  Mutiple short sessions will allow the youngster to develop good habits and a life-long ability to be properly handled.  Touch everywhere, even though you may never trim the nails, clean the ears, etc.  It is much better to touch everything and have them allow it, then to have a disease or need and have them fight you all the way. 
This isn't a complete list by any means, so stay tuned for more.