I've never seen a fuel gauge sender with three wires, but here's my two cents anyway. I believe one of the three is a ground. One may be for a low fuel light, and the third is for the gauge. TYPICALLY a fuel gauge is powered through the ignition/accessory circuit and indicates fuel level based on a variable resistance reading supplied by the float unit in the tank. I think we can all agree on that. As for diagnosing problems, to each his own. I usually start at the sender if I can get to it. Determine which wire is the sender wire and which is the ground(some systems are self grounded, others have a dedicated ground wire. Remove the sender wire, turn the key on, and "flash" the sender wire to ground. If the circuits are good, and the gauge works, the needle will move. Usually to full with a direct ground. If the gauge DOESN'T move, then trace the power to the gauge and or continuity from gauge to sender. In your case, since you never stated what, if anything, the gauge needle does with key on or off, that's hard to call from here. As for the low fuel light -- my guess is you have a third circuit for said light, with a contact on the float unit which simply completes a ground when the float drops to a certain point. Since you stated the light is on all the time, either the wire involved is grounded somewhere else, or the float lever is down all the time(float missing or full of fuel). If so, and all circuits do function, the gauge will read empty all the time.
