The first decision when choosing an animal to adopt is usually: dog or cat. Dogs are called "man's best friend" for good reason because they tend to be loyal, protective, and eager to please. But remember that they also need to be housebroken and exercised regularly. They may have "accidents" in the house and may shed more fur on your furniture. You may need a fence and a doghouse, and your new best friend may annoy the neighbors by barking.
Cats tend to be more aloof, independent, and nocturnal, bestowing their affection on their own timetable, sometimes at 3:00am. Some indoor cats never go outside. Cats generally use a litterbox instinctively, as long you keep it clean. Yet cats, too, will have "accidents" and some will use their main defense mechanism, their claws, to scratch or shred your furniture. Longhaired cats cough up hairballs around the house as well. And the mating habits of cats involve a lot of yowling.
Once you decide on a species, consider the age and breed you want. Choosing a puppy or kitten rather than an older animal lets you train your pet before bad habits set in, and youngsters are naturally more playful and fun. But an older animal, especially one that has been neglected or mistreated, will be very happy to find a good home and will probably enjoy a quiet, calm household.
Introduce your animal to its new home gradually, perhaps confining it to one room at first. Provide hiding places where your new companion can feel safe.
Different breeds have their own characteristics and temperaments, so consider whether you want a watchdog or a cuddly friend for your children. A big dog may live less than 10 years, while a cat may still be with you two decades from now. Decide who will be the primary caregiver, now and in the future, and look for a healthy, happy animal in either case.
Animals don't always understand or obey human rules, and communicating with them can sometimes be exasperating. Having a pet is similar in many ways to having children or caring for elderly parents. It is a 24-7 responsibility to provide food, fresh water, shelter, and medical treatment. But the companionship is well worth the time and effort.