I am a non-professional with limited electrical experience. I recently bought a house built in the mid-fifties that has some two-prong receptacles (i.e., no ground hole). I planned to replace the first outlet in the circuit with a gfci outlet, but when I turned off the power and removed the outlet from the box, I was surprised to find that the cable feeding it and the cable continuing to the next outlet are three-wire. So my question is: why would the person who did the original wiring not use a three-prong receptacle? How do I test to ensure the ground wire at the outlet box does in fact go to ground?
Connect the new receptacle with hot going to small slot, neutral to larger slot, "ground" wire to ground screw on receptacle. Turn power back on, insert a 3-prong circuit tester into the receptacle and see if the tester says you have a good ground.
Even if you don't have a good ground you can install a GFCI receptacle and get protection from any shock that a GFCI protects from (or at least some types of shocks).
If you find that the ground is not good then unless and until you fix this, just leave the new GFCI receptacle in place and put a sticker on the receptacle stating it is ungrounded. You can legally use it that way.
You could check in the main panel and see of the grounds return to there. Just measuring with an ohm meter neutral to ground Will tell you if they are connected but you would want to verify its not a boot leg ground (power off just in case) . a bootleg ground is when the neutral and ground are tied together outside the main service panel this is the only place they should be connected.
I would start with a physical inspection of all the wiring, starting at the panel, then following the cables as able, and inspect each junction box, to assure grounds are really there.
In the early days of grounding, a) the wires were grounded to the steel junction box, and b) the receptacle was grounded via its mounting screws. A) still applies and b) is illegal for receptacles now. If you've seen 3-prong cheaters that have a little spade designed to go on the coverplate screw, that was that.
Next, I would measure voltage between hot and ground to assure continuity, should be same or higher than as hot-neutral. If it's lower, like 93 volts, that is "phantom voltage" indicating a wire is disconnected between there and the panel.
Last, I would improperly connect a receptacle with ground going to the silver screw, and neutral unused. Plug a Kill-a-Watt into that, and a 1500W heater or hair dryer into the Kill-a-Watt. Set it for amps. Turn on the device for just long enough to get a stable amp reading, which should be 12.5 amps if voltage is 120V and proportionately less if voltage is less. (about the figure you get when it's connected properly). Then turn it off. That proves you have a ground path thick enough to carry fault current, which is the point of a ground. Disconnect it immediately, don't walk away from it wired wrong (except to turn off the break to disconnect it, obviously :)