The lamp just wouldn't light obviously.
There is also a separate socket inside to light the CMY numbers from the rear. This one is the same type used for running lights on older US cars.
Start with the power supply. Open the case and first look for any bare wires; especially bare wires close to the case. Vacuum out any dust and wipe dirt off any surfaces that you can get to. Replace bare wires or splice in better insulated wire. Clean out any corrosion. Clean all terminals and sockets. Others can suggest additional things to look for. I have not worked with electronics for a long time.
After the power supply, look at the enlarger head.
Then do the same with any item that causes the problem when attached.
Is your GFI part of an electrical panel and looks like any regular circuit breaker but with a "test" button or is it a type that looks more like a socket (except it has a test button) and installed on a wall?
You said you tried different socket in the house and it all did the same thing. Were all other circuit breakers GFI type also or did non-GFI type tripped as well?
Is your plug 3 prong type or 2 prong type? (in other words, is there a round ground pin?)
Here's something strange. Not all wall outlets in a typical home isn't GFI protected. Typically, kitchen, bathroom, and garage outlets are protected, but no where else. So please re-check your findings.
Assuming it IS really tripping the GFI that is installed on the wall and looks like an outlet...
What that means is, you likely have some "leakage" somewhere in your enlarger. You have two wires in A/C outlet. One is the supply line (black) and the other is the return line (white). All of the current from supply should return to return. Well, in your case, it doesn't.... some of it leaks to the chassis and your GFI breaker is sensing this. If it weren't for the 3 prong plug where one is ground, you may even feel a shock when you touch metal portion of your enlarger.
You are dealing with fairly old unit. Your transformer could have lost insulation. Some rodent could have chewed on wires. Maybe there's heavy dust on something which in turn got wet creating a pass for electrical power to leak.
As to what you can do, really depends on your skill level. What I suggest you do is to unplug from the wall. Let it sit for 30 minutes or so to let any charges bleed through. Then start your visual inspection. Start from your plug. Then the wire, all the way to the chassis. Any cracks, loss of insulation, etc, should be noted and repaired. Then, carefully open the cabinet and see if you see heavy accumulation of dust, loss of insulation, or any damage that are visible. Old transformers can leak and that damage may or may not be visible as it is an internal damage in the windings and insulation. If there is a plug, note which direction it plugs in and unplug it, then clean, then plug it back in. Do this one at a time.
Since you have no electrical experience, we are strictly looking for damages that you can see.
If you plug this into an outlet without a GFI, it will probably do something but I DO NOT RECOMMEND THIS. If the damage is substantial, you may be risking your safety. If you should do this CAREFULLY JUST TO TEST, and you find regular circuit breaker trips too, then you have a dead short somewhere and that's a whole different problem.
I'm going to stop here and see what you can find. Please make sure it is unplugged before you attempt any of this, and proceed slowly and carefully. Take notes on how you disassembled and unplugged anything.
I've actually got the enlarger in a studio, where all of the outlets were wired with a GFI switch. It was like that when I got the place.
I will definitely do as you suggested and go over the enlarger with a fine-toothed comb. I'll message back here what I find in the next couple of days. Thanks!
Only just dropped in on this query of yours. First simple test, the Chromegatrol will run without the six bladed head plug in it's socket, try this and if this does not trip the switch then your problem is in the head. if the switch is tripped then it's in the Chromegatrol setup. If it's the head get a professional to check the wiring as it does some loop back type stuff.
If it's in the control/transformer then google "ask harry" and follow his advice.