The washer doesn't spin or, if it does, it's real sluggish.

No spin at all: the lid switch is fried. What to do? You test it with your meter.
In the Whirlpool/Kenmore washers, a common problem is that the lid plunger stops making contact with the lid switch. Use a pen to manually press the lid switch actuator (with the lid up, duh). If the washer spins...well, you know the rest.
Older (belt-drive) Whirlpool/Kenmores: the spin solenoid is burned out or has cut wires. Ohm out that solenoid (20-30 ohms) and test with a test cord. Make sure the wires are intact by giving them a little tug. If this is OK, you may need to adjust/replace the basket drive.
Newer (direct-drive) Whirlpool/Kenmores: worn out direct drive coupler.
The drive belt has had it. Look for excessive glazing on the sides of the belt or cracks in the power side of the belt. On Maytags, replace the belt set if they look glazed or shiny on the sides even though the belts may look OK otherwise. Belts on other brands will be more obviously bad. Here's how to replace the belt on older Whirlpool/Kenmore belt-drive washers, and on old-style GE/Hotpoint washers.
Timer is fried. On older timers, it's sometimes possible to run an external jumper to replace the bad internal contacts. Usually, however, the entire timer must be replaced. The only way to confirm is to use your meter and wiring diagram.