No Picture or sound
The screen is blank with no raster at all. There are indications that the channel numbers are changing in the display. This indicates that some of the low voltages are present but these may be derived from the standby supply.
Assuming there is no deflection and no HV, you either have a low voltage power supply problem, bad startup circuit, or bad horizontal output transistor (HOT) or other bad parts in the horizontal deflection.
Check for bad fuses.
(If you have HV as indicated by static electricity on the front of the screen and you hear the high pitched whine of the horizontal deflection when it is turned on, then the following does not apply).
Use an ohmmeter to test the HOT for shorts. If it is bad, look for open fusable resistors or other fuses you did not catch.
Assuming it is good, measure the voltage on the collector-emitter of the HOT (this is safe if there is no deflection). You should see the B+ - probably between 100 and 150 V.
If there is no voltage, you have a low voltage power supply problem and/or you have not found all the bad/open parts.
If there is voltage and no deflection (no high pitched whine and no HV), you probably have a startup problem - all TVs need some kind of circuit to kick start the horizontal deflection until the auxiliary power outputs of the flyback are available. Some Zeniths use a simple multivibrator for this - a couple of transistors. Others power the horizontal osc. IC from a special line-derived voltage. The multivibrator type are sometimes designed to fail if someone keeps turning the set on and off (like kids playing) since the power rating is inadequate.
Test the transistors if it is that type with an ohmmeter. If one is shorted, you have a problem. The usual way a TV service person would test for startup problems is to inject a signal to the base of the HOT of about 15.75 kHz. If the TV then starts and runs once this signal is removed, the diagnosis is confirmed. This is risky - you can blow things up if not careful (including yourself). See the section: Bypassing the Startup Circuit for details.
If you hear the high pitched whine of the deflection and/or feel some static on the scree, confirm that the horizontal deflection and high voltage are working by adjusting the SCREEN control (probably on the flyback). If you can get a raster then your problem is probably in the video or chroma circuits, not the deflection or high voltage.