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.