When riding in your car with your pet, make sure they are safely restrained. Ideally this means in a crate; if your car will not accomodate a crate large enough for your dog, using a seatbelt/harness is an option. Crates are by far preferable however.
Just as your two year old should not sit in the front passenger seat next to you, neither should your pet! The same air bag that presents a danger to a child is no less threatening to a dog or cat. Uncrated pets should be restrained in the back seat; crates should be securely fastened so they do not become a projectile and should be placed ideally in an area that would be appropriate for a car seat. For smaller crates, you can just place them in the back seat and run the seat belt around them.
Here's my set up in the back of my van for my dog and equipment (of course, my dog typically spends at least an hour a day in the car with me and we travel to shows frequently on the weekends, so we are a little more "permanent" than my typical client. This is a very typical set up for those of us who show, though). The crate is on a raised platform made of pvc and plywood and covered with carpet, and both are attached to the metal rings on the floor of the van with heavy duty cargo straps to secure them in place. This is quite important; a few years ago a fellow agility competitor rolled her RV; her dogs, who were crated, all survived, but one of the crates was not secured, became a projectile, and struck her in the head killing her. So make sure everything is securely attached!
Last week I had a 3 1/4 pound chihuahua transferred to me from the emergency clinic. He had been traveling with his owner and riding in the backseat (good) but loose in the car (not so good, but I am guilty of this occasionally as well when I drive my mom's Mini!). She made a stop and got out of the car and the dog decided just as she started to shut the door to try and dart out from behind the back seat. His head was shut in the car door and he had some pretty serious resulting injuries. He spent the next three days transfering between us and the emergency clinic as he needed critical care and was lucky enough to survive, but will have some long term issues as a result of the accident.
Just two days later, a couple was traveling with their nine border collies in their Suburban from Pittsburgh to a show in the Columbus area. They were sideswiped by a semi resulting in a serious accident and the dogs all were loose on the highway. Someone had the presence of mind to call the show secretary, who put out a call for help on our agility email list and I understand that nearly 50 fellow competitors showed up to search for the dogs and all were found that same day. One of the owners however was not so lucky and sustained serious injuries. Most show competitors travel with their dogs crated, in this case due to the number of dogs in the vehicle it was not practical and it was only due to very good fortune and good friends that all the dogs were recovered.
When choosing a crate, the two main issues are ventilation vs protection. Plastic crates such as varikennels may provide more protection than some wire crates, but significantly less ventilation which can be a big factor when traveling in warm weather. For short trips and when the crate is not left in the care between trips, the ventilation is less critical. I use a wire crate both for my ability to see out my rear window and for ventilation, but I am less than satisfied with its sturdiness. I reinforced it with cable ties for strength to help, but I am still looking for one of the old style, heavy gauge wire crates similar to those with square mesh made by TF Scott, now out of business. Central metal makes a similar, but size options are very limited.
For those of us who do long trips and time frames in the car, protection from heat is essential. If you look closely at the pictures, you can see the fan on the back of Cory's crate; I have a golf cart battery powered generator which will run it for the better part of a day. You can also buy smaller vans that run off of the cigarette lighter attachment, but I like this bigger one as it puts out far more air. I also use a product called a "hatch latch" available from http://www.cleanrun.com/ which is a solid bar that attaches to the latch of the van hatch and allows me to leave it slightly open, but still locked to no one can get into the car. Attached to the front of the crate is the monitor for a product from Radio Shack called a wireless thermohygrometer, which has a remote reciever I can carry with me that transmits information about the temperature in the crate up to 300 feet away (far enough that I can run into the grocery for short stops and feel confident that I know if the car is starting to get too warm). Mesh aluminum tarps which block the sun but allow airflow can be attached to the van with heavy duty magnets or bungees, and are also available at cleanrun as well as many other sources. Between all of these, I can keep the van comfortable for several hours even on a day in the mid nineties; OF COURSE I check every few minutes even with these safeguards because nothing is more awful than a dog who has overheated in the car. These mostly allow me to make short stops on the way home from work, or to crate out of my car at shows for longer periods when the doors can be opened up for ventilation.I also try to keep a leash clipped to the front of the crate at all times in case the dog needs to be removed in a hurry; I also used to keep wire cutters attached to the crates just in case, but will admit they have disappeared and need to be replaced! Especially when traveling long distances it is important to have ID tags on your dog with a phone number where someone can be reached- ideally your cell and an alternate contact in case of an accident where you are incapacitated. For years I had emergency info laminated and attached to the front of my crates; I never added one for Cory but Andy's contained all the info. Since he passed away I've removed his crate and need to do a new sheet for Cory. I included their picture (for easy ID and in case they were lost I had one readily at hand to use to make flyers), age and health info, several emergency contacts both for the dogs and me, and a statement in red asking that in an emergency they be taken to a veterinary hospital or boarding kennel, not a shelter, and the numbers for someone who would guarantee their bill. I also included information on vaccine dates.
But probably the most important thing I've done to ensure my dogs ride safely is to teach them they are not to leave the car until I clip on a lead and give the ok. Even when we are at a show and Cory is at fever pitch and ready to explode he will wait in his crate with the door open until I am ready for him to get out. This can save a life if you are stuck by the side of the road on a busy highway. The best thing is, I taught Levi many years ago and have never had to teach it since! Levi taught Andy and Andy taught Cory- no way is the older dog going to allow that young whippersnapper to beat them out of the car- they make that clear from the beginning!
Hope I have given you some food for thought; even if your dog only makes the short trip to the vet once a year, if he is smaller beg or borrow a crate for safety and even Biggs is now carrying harness type seat belts. For cats it is especially important to confine them as they can escape very easily, and also can become frightened and bite their owners severely if they are held on a lap or loose in the car. No one ever thinks their cat will do this until it happens, but I am here to tell you ANY cat will react this way under the right circumstances, including my own. So err on the side of caution!