First simple transformer test is the "nose test" - if it smells burnt (the smell is rather
dis-stink-tive)
then it is likely the trannie is just a heavy metal weight. If it doesn't smell bad, then proceed on:
You can do a simple check of the power transformer without applying any AC voltages to it, with an ohmmeter, just check for the resistance between the primary leads (generally these are black, or one black, and one white) and this should be between 3 and 20 ohms, and not a dead short and not "open".
For the following procedure, I am making the presumption that the transformer leads have the "standard" US color codes used in that era.
The colors may be faded, but they should be close enough to discern.
Next check the resistance of the HV secondary winding, this has three (maybe four) leads, generally two red "high voltage" wires which should connect to the rectifier tube, pins 4 and 6, and a yellow/red striped, or perhaps orange, wire which connects to ground, generally at one of the ground tabs of the "can cap". A bias lead may be blue or gray.
Pull out the rectifier tube, and measure the resistance between pins 4 and 6, and then from ground to 4 and ground to 6. The resistance readings can be between 20 and 200 ohms, depending on the voltage and current ratings, and should not be shorted or "open".
The resistances most likely will not be identical, a 10 to 20% difference is normal, due to the winding's internal "layer wound", increasing diameter, structure, and one side will have more wire for the same number of turns.
Don't worry about teh filament winding resistance for now, as they are much less likely to fail before the primary or HV secondary windings.
If the can cap is bad (shorted) it would have most likely damaged the rectifier tube first, and if a properly rated fuse was on the primary side of the transformer, then simply blown the fuse, thusly protecting the transformer. If someone decided to insert a 15 Amp "no-blow" fuse into the fuseholder, then the trannie is most likely "toast". Big OOPS...
If you are nervous or uncertain about measuring potentially dangerous / lethal AC voltages, then get assistance from a friend or someone else who knows how to proceed safely.
Follow the "one hand rule", the black / ground meter lead should be clipped to the chassis, so it doesn't have to be touched. then use one hand only, to maneuver the red "hot" meter lead, and put your other hand into a pocket, that way the danger of an across-the heart electrical shock is reduced (never assume that it is fully eliminated).
Also, no bare feet on cement cellar floors, needless to say, and avoid any skin contact with the chassis or any other nearby metal object.
Also a good idea to remove any rings, watches, etc as these metal objects can, and will get in the way, in the worst way.
If everything looks good, then put in a low current fuse (1 or 2A) and remove the rectifier tube, and see if the other tubes light up (there will be no B+ voltage present, and perhaps a reather low and mostly harmless negative bias voltage will be present).
If this looks good, then physically connect (clip) the black lead of the voltmeter (set to its maximum AC voltage range ) to a convenient chassis "ground" point, and then, VERY CAREFULLY, measure the AC voltage which should be present on the rectifier tube socket pins 4 and 6, from the top side of the chassis, and keep in mind that the pin count goes counter-clockwise (to the left) from above, starting at the "notch" in the tube socket.
Without the rectifier tube, there should be only 5VAC on the pins 2 and 8, and this may or not read properly, in reference to the chassis ground.
The two other windings, 5V and 6.3VAC will read as very low resistance, under one ohm. Generally the 5V leads are yellow, and the 6.3VAC leads are green, and if there is a cventer tap, this wil be green /yellow striped.
If it has a bias tap on the HV secondary, this should generally be gray or blue, the normal AC operating voltage will be around 40 to 55 VAC.
HTH
/ed B in NC