Don't suppose any one would want to help me test this thing? I'm still learning and don't want to blow it up, or wind up killing my self probing something I should not probe. Better safe than sorry, right?
Certainly better safe than sorry, right.
It does take some degree of confidence to start prowling around inside an oscilloscope. The first things to check, as others have said, are the power supply voltages. Since you have a beam, you probably don't need to worry about the HV section itself (yet), and the low voltage supplies can be checked with a simple DMM. This isn't too risky. You will need to use an alligator clip to connect the negative lead of your voltmeter to the chassis ground. This keeps you from having to use both hands on probes; you will just carefully need to put the positive probe to the PS test points.
I haven't been able to find a _free_ copy of the 2213a Service Manual, and as I noted before, the front end of the 2213a is more similar to the 2215 model than to the 2213 model, so you will need both SMs eventually. Then you can follow the procedure in the 2213 manual to test your PSU voltages. The manuals are too large for me to attach them here but you can find them easily enough using Google.
Using the Tek SMs can be very intimidating if you haven't done it before, but the schematics are liberally scattered with test voltages indicated all along the paths, and it is often possible, with care, to track down faulty components on the boards using a DMM to check these voltages in a logical manner. However, this does take some skill and knowledge.
So the first thing is to get the manuals, study the appropriate sections (figure out how to locate the board illustrations and schematic sections for the Power Supply voltage test points and the Horizontal Amplifier, for example), and then decide whether you feel confident enough to tear into the thing physically, or even whether it is worth your while.
Also, posting the problem on Tek-specific websites just might turn up someone who has had the same problem and who can point you to a specific component or section that may have failed. This will cut the problem down considerably.
ETA: FWIW, here's the 2213 Operator's Manual, small enough to attach here but probably not helpful as far as this specific problem is concerned: