The rabies vaccine is required because it's protecting against a virus that can affect humans. Sadly, even though the other viruses are actually a high risk for animals (including wild species), they don't have the stigma that the rabies virus does and so they're left as the owner's choice. If Old Yeller had canine distemper, it might have been a different story.
Depending on where you live, you might only be at risk for certain viruses. For example, living in the north, I don't bother getting vaccinated for malaria, because it's a slim to none chance that I'll ever get malaria. You'll want to do some research, maybe consult a vet, to see what viruses are more prevalent in your part of the country, so you can weigh your options. Ultimately it's up to you, but I would like to point out that slight inconvenience now, could save you the inconvenience of a pet that contracts a painful illness later on. The costs of treating these illnesses are nothing to scoff at.
Like measles and the whooping cough for humans, there are some vaccines that every pet should have:
Core vaccines are considered vital to all dogs based on risk of
exposure, severity of disease or transmissibility to humans. Canine
parvovirus, distemper, canine hepatitis and rabies are considered core
vaccines by the Task Force.
Non-core vaccines are given depending on the dog’s exposure risk.
These include vaccines against Bordetella bronchiseptica, Borrelia
burgdorferi and Leptospira bacteria.
Core vaccines are considered vital to all cats and protect against
panleukopenia (feline distemper), feline calici virus [sic], feline herpes
virus type I (rhinotracheitis) and rabies.
Non-core vaccines are given depending on the cat's lifestyle; these
include vaccines for feline leukemia virus, Bordetella, Chylamydophila [sic]
felis and feline immunodeficiency virus.
These viruses are not just transferred from dog-to-dog or cat-to-cat. They also can be transferred through urine and feces. So while you might think your dog is safe not socializing with other people's pets, they are still at risk if you let them outside, or even if you bring it inside unawares.
Canine distemper especially is a big risk, because it infects all types of canines, and closely related animals as well so coyotes and foxes that might wander by at night when you don't see them could urinate and expose unvaccinated dogs who sniff that spot.
Canine distemper actually infected the black-foot ferret population, bringing them to what we thought was extinction. It's also thought to be a contributing factor in the Tasmanian tiger's extinction. I believe now it's also affecting African wild dogs, and even mutating to infect sea lions.
Canine parvovirus is deadly if contracted. The only way a dog can survive after catching this virus is through aggressive treatment in a hospital. The virus causes severe dehydration, and damage to the dog's intestines and bone marrow. Untreated the mortality rate is 90%, treated the mortality rate is down to 80%, so you can see why it's better to prevent it entirely. Not to mention it can infect puppies before they've even been born.
Canine hepatitis is a liver infection spread by all type of canines (wild and domesticated) as well as bears. It's spread through the feces and urine of infected animals as well as the blood and saliva. Luckily, this one isn't as deadly as the others.
Feline distemper is the same risk as Canine distemper (although without the history), it can be spread by other cats as well as minks and ferrets. Perhaps more importantly, it can be spread by fleas between all of the animals.
Feline calicivirus is one of the two virus responsible for viral respiratory infections in cats. It's about an even 50% of cats with respiratory infections have this virus I believe. The virus causes mouth ulcers and fever.
Feline herpes (rhinotracheitis), also known as feline influenza, is the other half of respiratory infections. In younger cats it can cause pneumonia, and supposedly it causes flat-chest syndrome but I don't think there's anything to support that.
In both the Feline calicivirus, and Feline influenza, cats can continue to carry the virus for years, if not for life.