Flavor, smoke point and nutrition are the three factors that influence which oil I use when cooking. The two oils I turn to first are olive and canola, because both are rich in monounsaturated fats. Regular olive oil has a mild flavor that pairs well with most Mediterranean foods, is highest in monounsaturated fats and has a smokepoint (the temperature at which the oil will break down) of 425-435F.
Canola has very little flavor, is lowest of any oil in saturated fat and has a high 435-445F smoke point. A good third choice is soybean oil—usually labeled "vegetable oil"—which accounts for three-fourths of all oil consumed in the US. It has no flavor and a smokepoint of 440-450F, a good balance of monounsaturated and polyunsaturated fat and is gentler on the wallet than olive and canola oils.
For more information on different types of cooking oils, see Cooking Oils 101.