It's probably not the blower.
In heat mode the furnace will light and run for a few seconds before the blower will even try to come on. If this is not happening then you probably have a control or power issue. There should be a low voltage transformer in the furnace, that is probably what you are hearing if it is buzzing all the time.. Sometimes they are just noisy.
Look for a control board and see if there are any lights on or flashing. Try heat, cool and fan-on modes, listen for any clicks from the unit. Listen for the outside condensor coming on when the AC mode is turned on. If you dont have any of this then you will need a schematic and a voltmeter (and know how to use it) to troubleshoot any further.
FYI, if you have a digital thermostat and it has a working display, there could be batteries in it. A working display does not mean it has power from the furnace.