Here are my notes thus far:

PROBLEMS:
-washer will fill with water
-agitator will not engage
-timer will go through the complete wash cycle but will not rinse or spin cycle

POSSIBLE CAUSES with NOTES ON INSPECTION
- bad belt *checked and operational
-bad drain pump * checked and all hoses secure, no blockages
- bad timer * checked/all connections look good
-electrical connection * checked and outlets are good

CONCLUSION:
I am led to believe after much Internet research that the problem lies with the TIMER

What are your thoughts and suggestions?
Thanks.

If nothing operates mechanically, it could be the motor. Do you hear it come on? Somewhere on the washer is a schematic of the wiring. Find the hot motor lead and jumper from the hot timer to the motor. If the motor comes on , it is a timer. If not, you probably have a bad motor or possibly a motor relay if so equipped.