My next door neighbor gave me a book to read - "Pet First Aid" by Bobbie Mammato, DVM, MPH. It is published by the American Red Cross. The book was reviewed by several veterinarians with excellent credentials. Everyone who has dogs or cats should have a copy for ready reference.
In many instances, the first question to be answered is, "Is this an emergency?"
The book provides a concise list of the common problems you are likely to encounter, and guidelines to help you answer this question. It has instructions for helping you check heartbeat and pulse, breathing, temperature, and checking for dehydration and shock.
Safely approaching and gaining control of an injured pet and transporting him to a veterinary hospital is the next challenge. There are sections on evaluation of risk, (to you and the animal), methods for capture and restraint, and techniques for transportation. You may make injuries to the pet worse by using improper capture, restraint and transportation methods, but more likely, you may suffer a serious bite or claw wound from your injured pet.
It is actually pretty rare that emergency treatment that you can do at home will make the difference between life and death. The ones that can be life saving almost always involve breathing or bleeding. The things you can do - clearing the airway so the animal can breathe, breathing for the animal if it's not breathing, and recognizing and controlling life threatening bleeding - are explained in understandable terms that you will likely be able to remember if you're ever faced with one of these emergencies.
There are good instructions for using chest compressions in cases where there is no heartbeat. There are instances where this may be worth trying, but I think it is rare that it saves a life. Even in the best veterinary emergency hospitals, the success rate is pretty low, unless the underlying problem that caused the heart to stop can be quickly fixed. It can also be very difficult even for a doctor with a stethoscope to be certain there is no heartbeat and in many instances squeezing on the chest of a seriously injured dog or cat with a weak, unstable heartbeat could do more harm than good. If your injured pet is unresponsive, not breathing, and you can't feel a heartbeat, "mouth to nose" respiration is certainly worth doing, and - careful chest compressions? - at least you will know you tried.
I also think it is rare that a tourniquet saves a pet's life, though it can certainly happen. Most fatal bleeding is internal bleeding, and most bleeding from external wounds can be adequately controlled by direct pressure. "Pet First Aid" has an excellent discussion of these things, but if you apply a tourniquet, be careful to follow the instructions exactly and get to a veterinarian as soon as possible. A tourniquet can do serious damage.
There is a very complete list of other emergencies and problems pet owners may wonder about, with easy to scan sections on the possible causes, the symptoms, first aid you can do, and preventative measures you can take. There are many things like minor abrasions, little cuts, broken toenails, and mild brief episodes of diarrhea that you may be able to handle at home. There are other things, like discovering that your pet has lapped up some antifreeze off the garage floor, that require immediate veterinary care even though the pet may seem perfectly healthy.
You can order a copy of "Pet First Aid" on the web at www.redcross.org or by calling the Red Cross at 1-800-667-2968.