Let your hosts know when to expect you.
When you head out for the day with your car, always let your hosts know where you expect to go and when you'll most likely be back. If you are going from place to place, let the proprietors of your next accommodation know when you'll be arriving and what route you will be taking. If you do get horribly lost or get stranded along the road, at least you'll know that one person has noticed that you've gone missing and they will have somewhat of an idea of where to start looking for you.

Make sure you have a spare.
In the US, it's easier to get help if you get a flat tire. Chances are you've got your cell phone on you and you may even be a AAA member, making it easy to arrange a tow. At the very least, you can call the rental company and ask for assistance. If you are traveling in another country without a cell, getting help is a bit more difficult. Always check to make sure your rental car has a spare tire, and before you set out on your trip, make sure you know how to change it.

Don't forget a map.
If you've got the cash and the option is available, get the GPS, but also bring a hard copy map with you as well. As we've seen, sometimes there's no substitute for an actual old-fashioned paper map. If GPS isn't an option, don't rely on vague directions, be sure to pick up a comprehensive map in case you decide to wander a bit or in the event that the directions you were given turn out to be less than accurate.

Know the rules of the road.
Stop at stop signs, don't speed, watch out for children and livestock. These are rules we know and which tend to be consistent across continents. Other rules of the road are more localized and often unwritten. Not following them may not get you a ticket, but they may not earn you any friends along the way either. Always research the road culture in a place you will be driving and learn customs that are followed there. For instance, when I was driving in South Africa, I was glad my friends had told me that on two-lane roads I should move over to the far left so that faster drivers could pass me. Had I not known, I probably would have made some other drivers very angry as they tried to pass me while I drove in the middle of my lane.

Don't make yourself a target.
If you are driving from place to place, you'll be traveling with your luggage and you may have a GPS unit mounted on your window or a map spread across the backseat. All of this screams "I'm a tourist, come pillage the car!" Always put your luggage in the trunk and stow the GPS and maps in the glovebox. Lock your doors when you aren't in the car and don't give anyone a reason to break in.

Read the fine print.
Be sure to familiarize yourself with your rental company's rules. Some don't allow rentals with debit cards, and a few countries require than the driver have not just a driver's license from their home country, but an international driving permit as well. If you're told something different in person than what you've read, be sure to ask for clarification. A couple I talked to in South Africa thought they needed to sign a special form to take their rental out of the country, but the rental agent said it wasn't necessary. When they hit a cow and totaled the car in Botswana, they were told that because they didn't sign the form before crossing the border, they could be liable for the cost of the car - about $7000US! Always read the fine print and know the rental rules.