The gray cable is one channel the black cable is the other channel. You would want to measure between the pink and white wire from each cable on the head. One will be hot and the other ground. The head has two coils in it. One for the left channel and one for the right. A coil is one long piece of wire wrapped into a coil. It should have low resistance. You have to check each one and compare. That means unsoldering one wire from the head and reading across the head with your meter. Red meter lead where white wire was hooked up and black where the pink is hooked up.Remember to do it with the pink and white from the same black or grey cable. The two coils are separated from each other. Or they should be. The brown is not rust, it is residue from the tape. The tape touches the head and will shed part of itself. Most heads are made of very hard material to be able to withstand the friction. I have heard of one set, I don't remember what brand, but it was too soft and would wear out quick. When it wears out, the tape starts rubbing the coil and will wear through the wires very quickly. This may be what has happened. The steel gap has worn until it is too wide and too thin and the tape is touching the wires. The best sound is attained by the smallest distance between the coils and the tape, but the wires are fragile and must not actually touch the tape. Your coils may have touched the tape. If the head is good, then I would put the meter's red wire on one end of the pink, and the black on the other end of the pink, and check the wire for a direct short. If it reads anything else the wire is bad. Check all four wires that way.
An OHM meter will put out a small voltage from it's internal battery, usually 1.5 volts on it's leads. It then measures the voltage back and calculates the voltage drop and therefore how much resistance in the circuit. A straight piece of wire doesn't drop much voltage at all and will read low on the meter. An open will not return voltage at all and will read infinity. On audiokarma, in the do it yourself forum they have a sticky at the top of the page for the N.E.E.T.S. manuals. Naval electric and electrical training manuals is what it stands for I believe. I ordered the set when I was in the Navy back many years ago. They will take you through the basics of electrical theory. They will explain the use of tools, simple circuits, and the basics of using them. I don't think their sets are complete but if you google neets you should find the complete set. They will explain a lot better than I can. Although, there are rules to troubleshooting. First you need to know how something is supposed to work, then you can see that it isn't working right. Second you need to know how it does what it does, then thirdly you can begin to come up with reasons why it's not doing what it should.