Any of the resistance or Ohm settings will work for continuity. I usually use the 1k (1000) ohm one but it doesn't matter. Some meters will have an audible signal on one setting but many do not. It's nice to have as you sometimes are not easily able to see the meter display but it's not necessary.
On any of the resistance settings, if you touch the probes together you will see the meter read zero, or at least very close to it. If this happens you have continuity. I always touch the probes together as a quick check as I start. You are now ready to go.
Touch one probe to one wire end, say the red wire at the headshell. The other one goes to the tip of the red plug. If you get "zero" or a beep you have a circuit, or continuity. Now touch any of the other rca connections. If you get zero/beep while still touching red, and only red, at the headshell you have a "short circuit" or "short" , for short (couldn't "resist"

). A short is a bad thing.
Test all 4 colours this way:
Red to center tip at RCA red
Green to outer "barrel" of red RCA
White to white RCA tip
Blue to white RCA barrel
Some cables will not use Red and White. They might have grey and black for example. This doesn't matter but what does matter is that one lead at the headshell connects with one and ONLY one of the terminals at the end of the cable.
You can also test from the solder lugs inside the turntable to either the tonearm end or the cable end. This will isolate the connection issue to that area. If it's a problem in the tonearm itself this is a bigger pain as it's trickier to replace the internal wiring than the RCA cable, but it is possible.