Boiled down to the basics, pet first aid is not that different from human first aid. But there are some big differences to be aware of:
#1: Dogs and cats (most of them, anyway!) have fur.
This makes wound care a bit more difficult. Generally, please don't try to use scissors on your pet's fur; even experienced groomers and vet staff can occasionally have accidents and cut the skin. It is usually fine to do an initial cleaning with warm water and/or hydrogen peroxide (alcohol stings, but is safe to use if tolerated) but leave further clipping and cleaning of a wound to your veterinarian, as soon as you can get to one.
Protecting the wound until it can be further treated makes sense to keep out bugs in the environment and to prevent self-trauma. Especially if there is some bleeding, covering with a temporary bandage or applying firm and steady pressure for at least 1 to 2 minutes, or longer for severe hemorrhage, will help speed clotting. Basic supplies for a bandage can be as simple as paper towels and scotch tape, in a pinch, or sterile gauze held in place by medical adhesive or elastic tape from your (hopefully well-stocked) first aid kit. The major thing to know about home or field-placed bandages is not to overly constrict what you are bandaging, as this can cause far worse complications than the initial injury (constriction leads to loss of blood flow and oxygen, and amputation in the most severe cases). And realize that what you put on will need to be taken off at the vet, so try to avoid the duct tape (but actually, if it is all you have, it gets the job done).
#2: Dogs and cats walk around barefoot, all the time ...
... so foot injuries, penetrating wounds, foreign bodies, infections and torn toenails are much more common for them than in people. Some of these are extremely painful, and your pet may not tolerate your touching anything close to the foot. Small animals instinctively guard their feet unless trained to allow handling for the most part, more so when they are painful. This would indicate you need to get your pet to a vet quickly to have him or her examined under proper physical or chemical restrain (i.e., sedation or anesthesia). This is sometimes not only more comfortable and less stressful for your pet, but may help prevent a lifelong fear of the vet clinic or of having their feet handled in general. (We all know a few dogs who have to have the groomer or veterinary staff trim their toenails because it takes a group effort of professional restrainers or sedation every time.)
#3: Dogs (and some cats) are indiscriminate eaters.
This means that anything, and I mean anything, is game for mouthing. We pull everything from sticks and other plant parts, underwear, knives from the birthday cake, fish hooks, tennis balls, porcupine quills, you name it, from their mouths and further down the GI tract (stomach and intestines).
Drooling, dropping food, or holding the head or neck positioned abnormally can be signs of something lodged in an odd corner of the mouth or throat. Items can migrate to some very interesting locations, which are not always straightforward to find. If the pet has swallowed something that passes into the stomach or somewhere along the intestinal tract, vomiting or pain and general malaise can follow. Inappetance (a decrease in appetite) to full-blown anorexia (refusal to eat anything) is common, as well as eating or drinking and throwing it up, whether after every meal or only some meals, sometimes right afterwards, sometimes hours later, and whether food appears digested or less so. Some will drink water but often not enough to meet hydration requirements, especially if they lose fluids through vomiting or diarrhea.
Edible foods also can make them quite ill (ever have a rough time following an outing to a spicy Indian or Mexican restaurant?). And don't forget food "poisoning" or ingestion of a large enough quantity of disease-causing bacteria that thrive in the environment of the intestines to overwhelm the immune system. Many of these produce toxins, the most severe and luckily not as common ones even causing toxic shutdown of multiple organs and death within 48 hours.
Inedible foods can be life-threatening, depending on the specific item’s potential to cause obstruction or other complications. And just because your pet has not done it before does not mean he or she will not do it for the first time, at any age.
#4: Last but most importantly, dogs and cats can’t tell us when something’s wrong.
It seems too obvious to need stating, but sometimes owners don’t understand why, as vets, we will focus in on a tiny detail of history that the owners give us. Sometimes we are extremely lucky to catch an early sign that something is wrong. If you notice any physical abnormality or sense something unusual in any aspect of your pet's behavior, it's best to trust your intuition and pay attention.