On what side of the furnace do I install the tank of the humidifier (the warm side or the return air side)?
All the humidifiers I've seen mount on the return plenum and have a duct installed that bypasses the air handler to the supply plenum.
Check with your humidifier vendor. Bypass style evaporative humidifiers can usually be installed on either the warm (supply) side or the return side; either side provides essentially the same results. Some prefer installing on the return side as, if it fails in a bad way, water is not dripping into the furnace / electronics leading to freezing kids / costly repairs.
Evaporative units with fans rather bypass lines normally go on the warm side however some vendors allow them to be installed on the return side provided hot rather than cold water is being used.
All others (steam, drum) based units, as far as I'm aware, attach to the warm side.
I've had two different whole-house humidifiers, both humidified the return (intake) side of the system, but in two different ways. One was a rotating disc system, which mounted on the bottom of the ceiling-mounted return; the discs rotated into the water and then up into the airflow. In our current house, the unit is mounted on a vertical surface of the hot side, and has a small pump that dribbles water down over a replaceable element; hot air from the supply side blows over the element it and back into the return duct through a small bypass duct.