Transistor oscillators are made up of an inductor or coil, a capacitor and a transistor. Usually there's other parts to make it all work, but that's the basics. In a BFO, two oscillators are tuned to almost the same frequency. they are offset in frequency so you hear the combination of them, or beat frequency.
One of the inductors is the search coil, the other inductor is inside the box.
Many modern digital multimeters (DMM) can read transistors and capacitors as well as resistance and voltage, etc. Testing capacitors is iffy, because they really need to be tested for leakage using the frequency at which the circuit operates. Most of the easy tests that a DMM will perform are at DC, so while they can indicate problems, they won't find all of the bad caps you come across.
You can read an electrolytic capacitor for simple DC resistance with a DMM. It will start out at a low Ohms reading and this will climb through several ranges on your meter. Eventually it will top out and hold some reading. The higher this is, the better, generally speaking. A very low reading that does not rise is probably shorted. One that has essentially no reading may be open, but not always. Swap the meter probes positive and negative, and the capacitor will discharge and then recharge and once again rise in resistance as it charges up again.
The other common test is for capacitance in microfarads (uf). Not all meters have this function. Most electrolytics have a broad range of acceptable values, so don't expect a "10 uf" cap to be exactly that, but if it's way off, that indicates a weak cap.
All this can be fun, but new caps are so cheap, it's almost as easy to simply replace them as to trust any tests you might try. At least one lead should be unsoldered to test accurately, so just remove and replace any suspect caps. If there's a lot of them, the tests might help narrow it down to the few bad ones, but you may have to repeat the process again later if you don't change out all of them. You also have to be careful not to create new problems when you do a lot of wholesale parts changing, so it's good to test along the way to see if it got fixed or became worse due to an error creeping in.
Sometimes caps will bulge or actually spew their waxy contents, but not always in the low-voltage circuits. You might see discolored or burnt parts, bad solder connections or physical breaks or it all may look just fine. But often close inspection can reveal the problem.
You can read the battery voltage with the unit turned off and then on. If the voltage really drops when turned on, that may indicate a shorted part. Usually the voltage should drop a little, just not too much. If it doesn't change at all when turned on and off, that may indicate a bad wire, switch, connector or blown part.
Transistors can fail either way, shorted or open. They can be tested like two back-to-back diodes or your meter may have an HFE test.
Mica "disc" caps are hard to test, but fortunately are usually still good. Don't just assume they are good, however. Resistors also change in value and can go open. Battery corrosion can travel a long ways up wires to other parts. Dirty controls and switches has already been mentioned, but don't forget the coil cable and coil itself as they can fail, too.
Play with your meter on known "good" parts to see what readings to expect. Sometimes you can just probe around and visually inspect things without having to unsolder parts. Take good closeup pics or draw sketches to show where wires go and how parts you remove are oriented for polarity, etc. Tiny wires break off as you move things around and without the pics or diagrams, you'll never know where they came from.
Don't wade in too deep to where it's too much of a mess for someone with more experience to be able to figure out the original problem, let alone any new ones you've accidentally created.
To become familiar with the techniques, you can try them out on some broken old radio instead of your detector. This is how I learned, and from reading the old hobby electronics magazines of the 1960's and '70s.
-Ed