You can eliminate the lid switch by seeing if it will spin. If it spins the lid switch is all right. If it won't spin or agitate, check the lid switch. If it spins but won't agitate, check the water level sensor. Look in the back panel, and you'll see a gadget that looks like a little canister, sort of like the front axle motor on a disco D30, with a tube leading up from the tub. If you're reasonably handy with electricity you can test further by bypassing the switch in this sensor to trick the machine into thinking the water level is up. If it agitates then, it's prettly likely the sensor or the tube to it (look for lime and cracks, etc.).
I should put in the usual disclaimer that if you're not comfortable working around lethal voltages while standing in a pool of water, etc. etc. blah blah, then call a repairman.
ETA: On my maytag, the lid switch will not allow the motor to run at all. It's how you can set it to soak for long periods - just open the lid during the fill. So if the motor runs when it's supposed to agitate, and it still does not agitate, then do suspect the transmission. I should also add, in case you hadn't understood, that the machine is set so that it will not begin agitating until the level switch tells it to, so a bad level switch will definitely prevent all agitation. However, it will always spin empty or full. The spin cycle is timed, not sensed.