What's wrong with my Kenmore model 110.20702990 washer when it will agitate but will not drain or go into spin?

If your washer will fill and agitate okay but won't drain or spin the most likely cause is a bad lid switch.

You don't give many details but if the washer is agitating then the motor is most likely okay. On this washer a bad lid switch will allow it to agitate but it will not allow the washer to drain or spin.

To access and check the 3949238 lid switch you have to remove the washer cabinet. The instructions to remove the cabinet are in the image below. Your particular washer has two end caps that must be removed to access the two screws holding the console down.

If the motor seems to be trying to run then it's possible something is stuck in the pump but with the cabinet removed