If you want to travel by car, your own is best, of course. Rentals from your hometown are fine, too, if you aren't going too far afield. But for driving somewhere more than about 500 miles from home, you should definitely consider fly/drive. Most airlines have fly/drive packages, often tied to resorts or hotels, so don't dismiss them just because they are "for tourists." You will often save money on such packages over what you would pay for the individual items put together on your own (airfare, car rental, hotel rooms and the like).
Be sure to check ahead before you rent a car, particularly if you plan to do so on arrival in a foreign country--you may find the rules are stacked against you. In several countries, you can't rent a car if you are too old (65 in some cases, 70 in others). In most places, you can't rent if you are too young (under 25 much of the time). Insurance is a problem to think about too, with several nations--Ireland, Italy and Israel among them--having more severe restrictions than most.
You'll thrill at the super smoothness of major European motorways, and--in a negative way--at the wild abandon with which local motorists press down on the gas pedal. Especially nerve-wracking in Europe are Italy and Turkey, though Greece is not far behind in the cavalier attitude of its drivers toward safety and courtesy. Don't even think of driving in India, where you compete with elephants, monkeys, people and sidewalk merchants for use of the road, day or night.
Driving on many Caribbean islands and in Ireland can be idyllic, so long as you take it easy, stopping at beaches or ruins, taking time for picnic lunches and fishing along the way.
There is much to be said for using North American-based rental car agencies, especially if you are unlucky enough to run into trouble and have to settle it on the spot. You can always continue the argument, if there is one, when you get home if you have booked with an agency in this country.
