I think your water level switch is stuck in the closed position. It is telling the timer that the machine is filled to the correct level of water and to start agitating. (I assume the same thing happens on the rinse cycle).

You should try replacing the water level switch. If you want to try to test it out and are not to picky, (I am not, but then I do some pretty gross things working on these machines anyway) you can take it off and suck on the plastic extension that the tube connects to. You may be to unstick it in this way. If you can, I would still replace it, as this will tell you that you have a high probability of sucess in doing so.

Here is the new part.

Part number: AP3019752