Look at some of the below issues to help troubleshoot your washer. Whirlpool washers are sold under several names, including Kenmore, Roper, Kitchenaid, and Estate. Many of the same tips apply to other brands such as Maytag, Amana, Frigidaire, Electrolux, GE (General Electric), Hotpoint, and many more. If a tip is brand specific, we will try to note this.

Washer won't fill

Check your water inlet valve, which supplies water into the machine
Check your water supply into the machine from the wall.

Check your pressure switch / water level switch and hose. (in this case, the machine would immediately start agitating even though it is empty.)
Check your lid switch. Some models such as older Maytag Washers have a lid switch that will not allow the machine to fill if the lid switch has failed.

Washer overfills

Check your pressure switch / water level switch and hose. Sometimes the hose is just clogged with debris or has a hole in it.
Check your water inlet valve, which may be sticking open and overfilling. If this is the case, the water will not stop filling even
if you turn off the machine.

Washer fills with water when off

Check your water inlet valve. If the valve is not shutting all the way, it will gradually drip water into the machine even when off. This is solved by replacing the valve.

Washer only fills with one temperature of water

Check your water inlet valve. This valve has two solenoids, and if one has failed, you will only get one temperature of water into the machine.
Check your temperature switch or timer. These are rarer to fail, but you can reverse the wires on the solenoids of the water valve to find out if the problem follows the side of the valve, or the electrical wires.

Washer will not agitate

Check the gears between your top and bottom half of the agitator (Whirlpool models)
Check the motor coupling between the motor and transmission (Whirlpool model)
Check the belt between the motor and drive pulley (belt driven models like GE, Hotpoint, Maytag, Speed Queen)
Check the lid switch. On some models this will even prevent agitation