What is a quick and easy way to numb up a part of my body such as my tongue or arm?

It just depends.: It depends on why the body part needs to be anesthetized. Dentists can numb the tongue easily with an injection. Anesthesiologists can use more complex techniques to anesthetize a larger part of the body like a hand, leg or foot.

Topical agents: Numbness is caused by disabling the sensory nerves that supply an area. In medicine, this is accomplished on a regional (arms or tongue) basis by injection of local anesthetic agents. The only other way to do this is to apply topical local anesthetic agents and this will only 'numb' the skin to which it is applied.

Apply ice.: Applying ice to the area for 5 minutes this will numb up your tongue or arm. The extreme cold inhibits your pain receptors for a very short period of time.