Your washer won't enter the spin cycle if the lid is open. Move any objects keeping the lid from closing, and shut the lid. The lid switch lets the washer control board know that the washer lid is closed. The water level switch determines how much water is in tub and whether that is enough water to start the cycle. If either of these switches is defective, your washing machine won't start spinning. Disconnect the washing machine from the outlet, and remove the main access panel. Test both switches with a multimeter, and determine if they have continuity. If not, get new switches from a washing machine parts dealer since it's difficult to repair these parts.
The timer advances the washing machine through the wash cycles. If the timer is not working properly, your washer may not enter the spin cycle. Unplug the washing machine, and test the timer with a multimeter. If it doesn't have continuity, you must get a new one since it's hard to fix the timer. Certain types of washing machines have motor couplers, which will be located between the transmission and motor shaft. These couplers easily break after repeated use since they are made of plastic and rubber. Replace the coupler.
Your washing machine won't spin if the transmission is leaking or has broken gears. Remove the main access panel, and examine the transmission. If the transmission is defective, you can rebuild it if you have the proper knowledge. If not, you must replace the transmission. If the transmission is leaking, check the clutch. If the clutch is oily or is worn out, get a new one since you usually can't fix this part.
The agitator connects to the transmission shaft. The connection can wear out, preventing the agitator from rotating correctly. If the agitator itself looks fine, you may only need to replace the transmission spline. If not, replace both components. Your washing machine has either one or two belts, which can break or slip off. Reattach the belt to the pulley if it's fallen off, or replace the belt if broken or extremely worn.