Now you hear everything engage, then the click . Usually after the click the burner's light. But now they don't.

Does it sound like the relay switch or what do think?

Could be the hot surface ignitor, if equipped with one, flame sensor, any number of safties, control board issue, gas valve. Need to watch what happens after inducer fan runs. Do you see any glow? Look at the schematic on the door and also look for flashing LED's on the control board. Get back to us with more info.