I'm not well versed in the pre '73 setups, but assuming they are the same/similar....
What does or does not the gauge do? Normally the first test is to find the sending unit wire going to the tank, disconnect it (the '73+ have a connector near the tank for this) and with the key on, note if the gauge needle goes past empty or past full. I never remember which is which, but it's enough to know that the gauge SHOULD peg one way with this test. If it does, then take that wire and ground it to the frame. Guage should peg the other way. GM wants you to wait something like 5 minutes, my experience is that it will sweep from full to empty in seconds, not minutes, but YMMV.
While you are under the truck looking at that wiring, check that the tank ground is good. Again '73+, there is a ground wire from the sending unit to the frame that tends to be problematic, normally disconnected or rusted at the frame. If that wire is bad, gauge will not read right.
If this test fails, then need to move on to the wiring and gauge cluster. Possible that both the sender AND wiring/gauge are bad, but not likely. These setups are mechanically pretty robust, and luckily are pretty simple to diagnose. Again, if grounding/opening the gauge sender wire doesn't solve the problem, then you can focus on the gauge/wiring up to that connection, and once that is fixed, that test will work. At that point you would know there is a sender issue as well if the gauge still doesn't work.
