If your furnace is making a humming noise when it's not being called to turn on and heat or cool then that's normal and is usually just the transformer humming.
But if your furnace is humming and not immediately starting the blower when the furnace or A/C starts up then it's likely your Run or Start Capacitor which is usually located on or near the blower motor.
Check to see if the capacitor is swelled out and if it is it's bad.
A capacitor can also be going bad even if it's not swelling.
The Start capacitor stores a high voltage charge to give the electric motor enough power to start spinning it and without the capacitor the motor would just hum.
The capacitors are pretty easy to replace and are not that costly but if you feel uncomfortable working with electricity then it's best to call in an HVAC company to replace the capacitor for you.
I've replaced a capacitor on my furnace before and it was just as simple as discharging the capacitor using a screw driver to touch both terminals at the same time and then removing one side of the wires at one time an plugging them into the new capacitor.
If you don't know what you're doing or don't think you can safely replace the capacitor then please have an HVAC company come in and do the job for you since the capacitor can store a deadly charge in it and can kill you.
Always turn off the electricity to the furnace at the breaker before working on the furnace.