Many door problems are caused by poor workmanship. If a door is out of level or out of plum, rehanging may be required. Check the floor all around the doorway. In this area the floor is often unlevel, but unless the problem is severe, there is no reason to be concerned.
In most homes, cold drafts from underneath doors are a common problem. Exterior doors and doors leading to an unheated areas such as a garage or basement should be inspected for gaps at the bottom. These gaps should be weather-stripped.
Air leaks at the tops and sides of exterior doors are usually not as critical as those at the bottom. But these leaks also waste fuel, also all sides of exterior doors should be weather-stripped.
Do the hinges squeak when you open the door? The solution may be simple – like oiling the hinges. Or, the doors maybe warped, placing a strain on the frame. Warped doors that won't close tightly and that are beyond the help of weatherstripping should be replaced.
If a replacement is needed, look for a prime door with insulation value as well as beauty. Doors with a polyurethane core insulate better than solid wood doors.
Look at the framework that surrounds the doors. Do you see any cracks? Cracks often indicate that the wood may not be in good condition. You obviously don't want trim work that is rotting or decaying.