Loosen old, corroded brass plugs with a hammer and cold chisel. Rent a power auger with at least 50 ft. of cable. Start by removing the strainer that covers the drain hole. Then, look for a clean-out plug on the side of the drain basin. That allows you to bypass the trap and feed the cable directly down the pipe.
Some models come with interchangeable cutting blades.
Plug in the power auger and position it near the drain. Most models are fitted with a foot-pedal switch, leaving both of your hands free to guide the cable. Feed several feet of cable down the drainpipe. Push the cable into the pipe until you feel resistance or hear the motor start to bog down. Stop the motor, reverse the rotation and back out a few feet of cable.