I've been posting since yesterday about my emergency furnace replacement, but basically I had very similar symptoms on my 18 year old Amana.

When I was trying to self-diagnose before calling in the professionals, I found there weren't that many parts in a basic furnace before the burners ignite:
-"fan on" is directly switched by the thermostat, doesn't help diagnose the heating components that go wrong.
-the furnace "humming", if it is softer than the "fan on", is probably the sound of the smaller draft inducer motor rather than the main blower. If on at least the thermostat is working and calling for heat.
-if the draft inducer motor is bad, as mine was, it won't activate the pressure switch. The pressure switch will then not let the electric ignition activate. The pressure switch could also be a source of failure, although aside from a leaky rubber hose, didn't seem like there's much to break here.
-the electronic ignition seemed to have two wearable parts worth checking out, a probe sensor that could get dirty and a little heating element that could crack after 3-5 years. I was able to rule these out on mine because it did fire up briefly once while I was troubleshooting.
-the gas valve attached to the ignition seems to have a lot of fail-safes that will shut off the gas within a few seconds if the ignition sequence doesn't proceed smoothly, so I think the chances of leaking gas as slim (but I'm no expert)

If you get past the ignition sequence and the burners come on, there's other things that can fail and shut it down, but I never got to those things. "the fan" that you mentioned will not come on until the burners ignite, so your problem may also lie in one of the areas above...

Had I known where to get a replacement draft inducer motor quickly (I don't), it looked like six simple screws would have swapped it out in five minutes. However, once I called in the repair guy, I felt obliged to pay him the $350-450 (my estimate of the labor portion of the $600 quote), or as I decided, to forgo the repair and apply the money to a new furnace.

If you put the thermostat on "Fan" does heated air come out? If so, it could be the fan limit switch or if equipped, a fan relay.