If the washer will fill and drain, but won't agitate or spin, you may have a broken motor coupling. The noise of the motor engaging, lack of agitation and ability to still drain are the usual symptoms. This is a common fail item with this model washer through normal wear and tear. Newer versions of the coupler are more resilient and resist breaking. The following link explains how to replace one:http://www.fixya.com/support/r385250-replacing_motor_coupling_kenmore_top
The motor coupling is located between the drive motor and the transmission and takes the place of a drive belt. This configuration is known as a "direct-drive" system. They do wear out and can break. The replacement part number is included in the link I provided and is an improved upgraded coupling that is more resilient. It should last longer.
A replacement coupler can be purchased on line at appliancepartspros.com for about $11. Just use the part number in the link and "Search by Part Number" for your search criteria.
If the washer fills and doesn't do anything, or stops right after the wash cycle and then does not advance to the rinse cycle, this is commonly caused by a lid switch malfunction. The washer usually will NOT drain either. The lid switch is usually mounted under the washer top casing, on the right hand side on most top load models. It is activated by a plastic piece on the lid (called a "Lid Strike") through a small hole on the rim of the wash tub opening. In many cases the lid strike breaks off the lid, or the lid switch becomes loose and no longer makes contact. Inspect the lid to make sure the lid strike is still intact, and then inspect the mounting screws (adjacent to the small opening on the wash tub rim) and make sure they are snug. If you determine the switch to be defective, the following link explains how to replace one:http://www.fixya.com/support/r385198-replacing_lid_switch
On some models, the lid switch is located under the top casing by one of the lid hinges. You will know if you if you have this type of switch, because It does not have a lid strike and the small hole on the right hand side of the wash tub will not be present. This switch is actuated (toggled) by a rod on the lid hinge. With this type of switch, if you locate it and it can be toggled by hand, you may only need to adjust the actuator by bending it slightly.
A replacement lid switch (if required) can be purchased at any of the following web sites:
The average price is about $20 - $30. Prices differ between these sites, so shop and compare.
Take note of your symptoms:
1. Washer fills, may or may not agitate, will not advance to the rinse cycle, and will NOT drain - lid switch.
2. Washer fills, does NOT agitate or spin, but WILL drain - motor coupling.
If you still have questions, please let me know. If the suggestions I have listed do not match your washer symptoms, please post back and elaborate more on what the washer is/is not doing so I can provide you with better assistance. I hope this information is helpful to you.