No Fill?
I'm reading out of the book here, so give me a break. If you have your schematic, try and follow it from the power supply through the tests below.
Is there power at the outlet? Is the water on? I know. Simple, but it happens. Is there 120V where the hot and common connect to the timer?
Open and close the lid. Is there continuity through the lid switch when the lid's closed? Does the plunger and bracket assembly depress the lid switch button until it clicks when the lid closes? If not, loosen the switch and slide it forward until it clicks, then tighten. Check continuity again.
Set the timer to the middle of the wash cycle. Close lid. Pull out the knob. Is the solenoid getting 120 volt supply? Try another cycle like Permanent Press or rinse. Does the timer supply voltage to the solenoids when you pull out the knob? Do the solenoids in the water valve make any noise when knob's pulled out; like a click and buzz?
Disconnect the lines from the timer to the water valve. Apply a live test cord to the solenoids. They should click and then hum or maybe buzz. Is the valve clogged with debris? Clean screens.
Check the temperature switch. Are the buttons fully depressed? Is there continuity through the switch on the different temp. selections? Usually tested on opposite sides of switch, I believe.
Next try taking the hose off the pressure switch(water level switch)and see if you can blow through it, as Ben suggested. If it's clogged it won't work. If it's not then you can take a little hose that will fit on the pressure switch or another air dome and hose(if you have one)and attach it to the pressure switch. Blow in it until you hear a click. That's the switch closing, telling the timer to turn off the water and start the agitator. If you don't hear the click the switch is bad or stuck. If it clicks and doesn't stay clicked the diaphragm has gone bad.
If it clicks, pinch the hose shut with a pair of Vice-grips. If it holds air the washer will agitate w/o water if the timer's ok. Turn the timer to the middle of the wash cycle and pull out the knob. Try it on rinse and permanent press too. It should dry agitate in all wash settings. Remove the hose and reconnect the original hose that connects to the tub.
Push the buttons on the water level switch. The metal bar should move the center adjustment in an out when different buttons are pushed. If it's hanging you can try and lubricate it.
Start at the beginning and when you get to a test that doesn't work, suspect that part and double check. If parts don't seem bad check continuity through wiring. Look for burned or loose connections and broken wiring.
You probably know a lot of these things already. Just going through the list. If you get shocked, don't blame me.