What is the quickest form of heat transfer?

The quickest form would be through the transfer of energy in the form of electromagnetic radiation. Infrared Heating or Microwave Energy are examples of this. The radiation travels at the speed of light until the energy is absorbed in a material and you feel it as heat.

I accidentally hit the "Enter" key and my response got posted before I finished. Two other heat transfer methods are also noteworthy. In materials, sapphire is an excellent conductor of heat, even though it is an insulator. Silver is among the best conductors of heat for metals. However, the champ among materials is superfluid liquid helium @ 1.7 degrees kelvin.