Well....
Really, CRT is not something you want to mess with. Being consumer goods, many manufacturers take all sorts of shortcuts. It may not vaporize an entire tip of screw drivers but it may leave a mark. It may not kill you but you'll feel it for quite sometime. I have a test lead with melted part, and memory of burn (from 500v no less) to prove that.
Having said that, when people say measure voltage, they usually mean power supplies. At minimum, CRTs have anode, and heater. Logic and analog control has 5 volts and 12. Usually a few more for driving grid and deflection coils. I'd check all of them. Anode will require high voltage probes. If I have to place a wager, I'd say power supply is a good bet. You should at least do a visual check. Look for burned components, burned trace, bad soldering joints, etc. Look CLOSELY for discolored registers, bulged, leaking, or otherwise look funny capacitors.
Since you are determined to try this, I won't try to stop you. Just be careful. Check and double check power and large capacitors before you touch anything. For safety, use screw driver and discharge everything. It's not that dangerous if you know what NOT to do. But everybody is careless one time or another.
I have a friend who touched 3000 volts high current supply and lived to tell about it. Myself, I *almost* got myself few times. Good luck. By the way, people who tell you scary stories are really trying to help you, not discouraging you for any other purposes than to possibly help you save yourself from aggravations.
I have a habbit of saying outloud, "POWER OFF", "Voltage checked" when I approach anywhere near oscilloscope's tube section. Then make a point to take my free hand, and hit small of my back. I really don't like pain.