Besides the serial number, a TO30 has Z129 cast into the lower left side of the block and a TO20 would have Z120. TO30 oil filter is an external canister on the lower right side of the block and TO20s oil filter is inside the oil pan. The TO30 has an air intake opening in the upper right corner of the dash and the TO20 does not.
For starting, the shift lever activates a switch under the battery tray in the top of the bell housing. You can loosen the switch mounting plate bolts and slide it forward and back a bit to see if that helps as it may be out of adjustment. Also make sure all your wiring connections are clean and tight. The switch can be replaced if it is bad. You can test the starter by running a jumper cable from the battery and briefly touching the wire stud on the starter BUT be VERY careful doing this. The tractor must be out of gear and turn off the key so it doesn't start and run you over! If the starter spins when you jumper directly to it, you know its either your wiring or a bad switch causing the problem.
The TO20 and TO30 came standard with only the four speed transmission but there where aftermarket step up/step down gearboxes that could be added to get more ranges like the Sherman. Most likely you don't have one installed but look for a lever coming out the left side of the transmission down by the clutch pedal. Starting with the Ferguson TO35, a three speed with high/low range transmission became standard with a much greater range of speeds.