Add a c-wire to 25+ year old Rheem furnace
Based on your description of the wiring, and the schematic you've provided, this is what I think is going on.


Click for larger view

It sounds like there's a white wire and a black wire, connected to the secondary side of the transformer. The white wire carries 24VAC to the thermostat, which means the black wire is connected to the C terminal. 

So the new thermostat should be wired as follows:


White -> R
Red -> W
Extra (blue in diagram) -> C


You can verify this by measuring the voltage between the blue and white wires, in the diagram above. You should get ~24 volts AC, though it might be a bit higher or lower.