Set it to the lowest range for ohms. Put positive on one end of wire, negative at the other. If it reads 0 (or close) ohms, the wire is continuous all the way through between your test leads. If not, well, sometimes it helps to reset and try again and sometimes there's a wire broke inside the insulation. If you have a third arm or can use alligator clips to attach the test leads to the wires you can wiggle the mic cable and see if the connection comes and goes. You just have to make sure the test leads don't touch.
To figure out which wires are for making the relay go click, put one test lead on each of two wires. Press the PTT switch. If you get 0 ohms, then use those two. If one is a shield for another wire, that's usually grounded and the other wire in the pair will be the TX line. The wire left over is the audio line. If you have more then one wire left over then you have a spare wire, you just need to figure out which one it is.
If you get 0 ohms when the PTT is not pressed, then either the mic is wired for electronic switching and you've found the RX line, or it's got a serious problem. Since this is for a Cobra 138, it better not be wired for electronic switching.
You can also look at the PTT switch. By using the meter you can figure out which are the normally open and normally closed contacts, although it should be pretty obvious by where the wires are attached. I don't have a 138 stock mic handy, but I'm guessing it has a small DPDT switch in it where only four out of six contacts are wired up. Two of those contacts will be for audio, and two for making the relay go click. It can get confusing because the return wire for the mic may be wired to the shield or to the TX line using one of the contacts on the PTT switch. When the PTT button is pressed, it all gets shorted to ground anyways. Usually the "hot" side of the mic is wired through the PTT such that the mic element is only really connected to the radio's audio input when the PTT button is pressed.
Sorry, I don't have any real pictures to share. It would make this much easier. I did try some ASCII art, it didn't go well.
Please note that this is only valid for the 138 stock mic with the caveat that I've never owned or worked on one. Someone here with superior knowledge to mine may chime in. Listen to them, not to me.