• Pilot won't stay lit. Hold down the button, light, hold for a minute, let up. If the pilot goes out, it's the thermocouple. This is a $7 part.
• Pilot stays on but the furnace doesn't come on, step 1. Do you hear a click when you raise the thermostat setting? If no, the transformer failed. This is a $15 part.
• Pilot stays on but the furnace doesn't come on, step 2. Push the reset button on the limit switch and try raising the thermostat setting again. The limit switches on older homes don't have a reset button.
• Pilot stays on but the furnace doesn't come on, step 3. Replace the limit switche(s). You MUST know the brand and model number of the furnace to get the proper replacements. It will be helpful if you have the part number of the switches.
• Pilot stays lit, fire comes on, but the fan doesn't come on. Possible the fan switch. This is an $8 part located on the lower side of the furnace by the fire box. There is no way to test it, you just have to replace it and see if that fixes the problem.
• New fan switch doesn't fix the problem. That leaves the motor.
• Still doesn't work? Assuming you have fuel, this would indicate a problem with the gas valve. It's time to call in a professional; gas valve work is to dangerous for the inexperienced to attempt.

More about motors. Motors sometimes fail slowly. As they reach the end of their life, various parts start to fail so the motor has to work harder to turn the fan. This creates more heat which eventually causes the motors thermal overload protector to shut it off. After 20-30 minutes the motor cools enough to run again. It may take a month or two for it to fail completely. If that is long enough to get you out of the heating season perhaps you can postpone the expense of a new motor until the next fall. The risk you take is that the motor will fail completely some cold night and you will be looking at an emergency service call.