Washing machine will fill but
This assumes that you have a basic Kenmore / Whirlpool top load, direct drive, neutral drain (it doesn't spin until the pump removes most of the water) machine.
First, see the Sears parts site for your washer:
http://www.searspartsdirect.com/partsdirect/index.action
Enter your model number (###.#######) and you'll see a list of major sub-components with diagrams and parts.
Three possibilities: Ranging from easy to relatively messy but inexpensive.
Kenmore / Whirlpool top load, direct drive washers use a "Motor Coupler" between the motor and transmission.
It's used as a fail safe device and consists of three parts.
Two of the parts are the same, plastic units with three prongs that fit on one of the motor output shafts and the other on the transmission input shaft. Between them is a rubber piece with six holes.
The power from the motor is transmitted through the rubber piece to the transmission.
The other shaft on the motor directly drives the pump.
If the motor coupler is broken, or worn, the motor may not be able to send full power to the transmission for agitating or spinning.
See the following for how to remove the cabinet, pump, motor and motor mounting plate to inspect and replace the motor coupler.
http://www.american-appliance.com/catalog/newsdesk_info.php?newsPath=16&newsdesk_id=137
If the coupler looks damaged, it's a relatively easy fix.
Pry the plastic pieces from the motor and transmission. Some people use a claw hammer but I've found that using a small (6 inch) pry bar works well.
Carefully fit the plastic pieces over the motor and transmission shafts and press them in place (Use a 1/2" of larger socket to put pressure equally around the pieces.
Put the rubber piece on the transmission plastic piece.
Replace the motor mounting plate.
Carefully line up the plastic prongs on the motor based plastic piece to the three un-used holes in the rubber piece. If they don't line up right, the coupler could be damaged.
Mount the motor, don't forget the two screws, put the pump back on the motor and use its' clips.
Put the cabinet back on, don't forget to plug the lid switch back in and re-mount the console.
If the motor coupler is OK, the problem could be the clutch:
Sears has a replacement clutch band kit.
See the following for how to pull the pump, motor and transmission.
You DON'T HAVE TO PULL THE AGITATORS OR TUB.
http://www.american-appliance.com/catalog/newsdesk_info.php?newsPath=16&newsdesk_id=137&osCsid=65fc1c013d6130868b14c8bb4baf7bfd
Also see the Whirlpool / kenmore service manual for these washers.
https://secured.whirlpool.com/Service/SrvTechAdm.nsf/2cd44500d572193285256a45004fd9d6/f7e2147fa98b28cf85257122007475cb/$FILE/L-55.pdf
Both the Motor coupler (available from most appliance parts stores) and the clutch band kit are relatively inexpensive.
See the Following Whirlpool parts list for alternatives to the Sears part numbers.
http://shared.whirlpoolcorp.com/assets/pdfs/literature/Repair%20Part%20List%20-%20W10114743.pdf
It's also possible that the Neutral Drain Assembly in the transmission is broken:
The Kenmore / Whirlpool top load uses a reversing motor to initiate the pump for draining and the transmission to spin.
If it will drain but won't spin the problem is likely in the transmission. "Neutral Drain Assembly".
See the Whirlpool / Kenmore Service manual for top load direct drive washers.
WHIRLPOOL TRANSMISSION DESCRIPTION AND SPRINGS
https://secured.whirlpool.com/Service/SrvTechAdm.nsf/2cd44500d572193285256a45004fd9d6/2550f737b3129b0c85256b0600504738/$FILE/787772%20%234.pdf
Those cams/gears etc. in that assembly keep the washer from spinning until all of the water is pumped out and then go into spin mode.
Ours had a similar problem.
After looking at the transmission parts list (available on a number of sites) and on the Whirlpool site (Whirlpool makes Kenmore) we noticed that the one of the cams in the neutral assembly which is supposed to have a small spring, didn't on ours when we tore the transmission apart again
http://shared.whirlpoolcorp.com/assets/pdfs/literature/Repair%20Part%20List%20-%20W10114743.pdf
Found the part number for a kit (Whirlpool part number 388253 Neutral Assembly) for $15.00 and replaced the cams and gears.
In order to get the transmission out, you have to remove the cabinet, the fabric dispenser and both agitators. The top agitator uses a plastic 1/2" drive bolt (a 1/2" socket extension works well).
You don't have to remove the drum.
http://www.american-appliance.com/catalog/newsdesk_info.php?newsPath=16&newsdesk_id=137
Lay the machine down (after disconnecting power and hoses, remove the pump (two clips), the motor (two screws hold two clips on the motor), lift and set the motor aside.
Remove two 1/2 inch bolts for the motor mounting plate and three 1/2 inch bolts for the transmission.
Pull the transmission, remove the clutch (there is a spring clip that holds the clutch on the main agitator shaft), remove the 8 screws carefully the transmission holds 15Oz of 80-90 weight gear oil, remove the clip holding the main spin gear on and check the plastic cams.
If the cam that should have a spring doesn't, or the others look damaged, or the main spin gear is damaged the washer won't go into spin mode.
If that's the problem, install a Neutral drain kit, put everything back together carefully and the problem may be solved.
All of the above sound more difficult than they really are. If you're comfortable with a socket set, pliers/vice grips and willing to get a bit dirty (oil) you can save some considerable money.