Fixing old CRT oscilloscopes... what to look for first?
Get ahold of the service manuals first. Make sure you are experienced enough to be comfortable working safely with multi-kV voltages such as all CRT scopes use. There is quite enough voltage and current capability in a CRT scope to be lethal. If not, then stop now.
Check all the power supplies- both voltage and ripple, electrolytic caps die with age, particularly the ones in the power supplies that see a lot of heat and ripple current. Measure the ESR of the electrolytic power supply caps and compare with expected values. Most older scopes will not have low-Z caps so the ESR might be a few ohms. You can usually do a quick check in-circuit if the cap is discharged.
Some- the ones that use hybrids, for example, may not be realistically repairable.
Most parts, other than capacitors, don't die from age, at least not very quickly, and are just about as good at 50 years old as when they were new.
Beyond that, look at the waveforms in the service manuals, and see where they deviate. If the deviation isn't much out of spec, forget about it until you you find and fix the real culprit.
Keep in mind that failures can snowball- a failed fuse is sometimes just a failed fuse, but often there are several other parts that failed and the fuse just protected the wiring.
Inject signals at the inputs and follow them through the circuit once you get a trace etc.
I once fixed a scope that had resistors that had shifted in value by a huge amount (they had high DC voltage across them and were 1% semi-precision parts).
Oh, and if it becomes apparent that someone has already had a try at fixing it, you might want to leave that one for last or for the electronics recycling bin.
I can't really see a transformer just going bad, because you'd have to somehow manage to break one of the windings in it, correct?
If moisture gets into the windings it can corrode the wires or break down the insulation. This doesn't usually happen in normal operation because heat keeps the transformer dry, but if stored for a long time in a damp place it's a different story. Oscilloscope transformers usually have a winding that produces over 1kV, so they are especially vulnerable.
About half of the ones I run across and buy don't have traces... I'd at least like to get a trace on them. So what do I look for first?
Look for heater glow and a high negative voltage on the tube Cathode. Then check the grid voltage (controlled by brightness knob) and plate deflection voltages. Use a good scope to check the drive waveforms. If possible get a schematic, or infer it from a similar model. Also get the datasheet for the tube (or a similar one).
If the tube heater works and electrode voltages are correct then you should get a trace of some kind. Watch out for lack of horizontal sweep, as a bright stationary spot may quickly burn the phosphor.
Consider that most faults are not that complicated. Switch contacts go dirty and mess up amplifier biasing, making the trace go off-screen. Dry joints often occur on large components and parts that get hot. Electrolytic capacitors burst and leak over the PCB etc. Many of these faults can be found by physical inspection.
A few months ago my scope decided - without warning - to just put a dot in the middle of the screen. Luckily I noticed it right away, but after stripping the whole thing down I couldn't find the cause, and on powering up again the fault had 'cleared'. Of course it happened again a few weeks later, so this time I inspected the main PCB with a magnifying glass. Sure enough I found several dry joints on the connector from the power supply board. Before finding this simple fault I was seriously considering buying a new scope!