Start by finding out if you're lucky enough to have all the keys to the outside compartments, and then go look in them just for the heck of it.
Gently pull your batteries out onto the lid/cover and see if they have any fluid in them. Add distilled water (from any drug store) if they need to be topped off to the split ring.
Then put them back in their holes and plug the trailer in. Walk to the front and listen for a BUZZ. If you hear one, congratulations, you have a UNIVOLT. If you don't hear one, go inside and try some lights. If they work, even more congratulations are in order because you don't have a Univolt, which means that somebody put in something quiter. Then again, your Univolt could be both original and kaput. No lights at all would indicate that.
Then go to Wal-mart and look in the RV section. For less than $10, pick yourself up a little brass Marshall water pressure regulator. Put this on your spigot and then hook the water line to the trailer. Open the valve and count to five. Then turn it back off and go look to see if anything is leaking outside the trailer.
Then open the water heater door and see if anything is dripping there.
Then go inside and look under the sinks, in the water distribution center (which I think is in the cabinet in the bathroom in a rear-bath model), and around le toilette.
If no floods are apparent, go outside and turn the water on. Then hop back inside and do the inspection for leaks again. Listen for something that sounds like "Phssst!" That's bad news.
Then open the hot and cold water on the galley (kitchen) sink, and see if, after much popping and spluttering, you actually get water. That's a very good sign.
Try the same in the bathroom.
Under the sinks, you can see some water lines. Check them for moisture by feel.
Once you have either determined that nothing is leaking or have made note of the places that will need repairs, it's time to try the propane system.
Each appliance has a shut-off valve under the trailer. There's a good chance that these valves have never been moved. Crawl under there and find out if they will move. Inline with the pipes is on. Ninety degrees in relation to the pipes is off.
Try turning off everything, and then turn the gas on at the tanks up front. If you don't hear a hiss or hiss and rattle as the system pressurizes, then you have no gas. If the selector built in to the regulator is manual, slide it to other tank and open that one. If still no hiss/rattle, close the valves, take the tanks off and go to the propane store for filling.
This is also fraught with the potential for additional cost if the tank outlets haven't been updated. Your propane guy at the propane store will let you know if this applies.
Back at the trailer, slowly open a tank valve and listen for the hiss/rattle. If you hear it, great.
Then get your soap solution (thick dish detergent in a little water) and go around and check all the joints for leakage by slobbering the mixture all over the joints, which will make little bubbles if they're leaking. If you hear a big leak, skip the pressure test and turn the gas off.
Inline valves for seventies models can be had at Lowes. Be sure to use TWO wrenches when disassembling and reassembling.
When you have that marvelously exposed propane system repaired and repressurized, open the valve going to the stove, and then go inside and look at your stove. Is it a four-burner Magic Chef? If so, this stove has TWO pilot lights. One is under the stove top and is below the lower shelf near the rear of the burner.
To test, you don't need the pilot lights.
The left-hand knob should be in the fully clockwise position - NOT the OFF position because that would be too logical. (Push in and turn clockwise.)
Now take your camp igniter (long snouted lighter) and turn on one of the burners. Spark around the burner until you get propane to the stove. When it lights, do the same for the other three.
Now for the water heater. There are a few types. If you're really lucky, you'll have an extra switch in a cabinet or somewhere else in the bathroom that turns on the water heater. When you depress it, it should turn red to indicate that it's trying to light. If you don't hear it light, try it again. It may take several tries to get a startup.
If you can't find the button, go outside and lower the water heater cover. Do you see a potted circuit board and a big ignition wire going toward the burner?
If so, go back and look for that switch again. It should be in there SOMEWHERE.
If not, you have a pilot model. If you're really lucky, the instructions will still be stuck to the water heater. Try doing what they say, again using your camp igniter.
If it won't light, do a forum search for "water heaters" and you'll learn about propane-loving micro-spiders and other problems water heaters can have. Be prepared to be reading for a while.
Once you have the water heater lighted, go back into the trailer and open the freezer compartment. Do you feel any cold?
If so, great. Check it tomorrow to see how cold it gets.
If not, get on your knees and look at the fridge knobs. One should say "Electric." The other one should be about on number 4.
Well, this should get you started. If you plan to actually move it somewhere, there will be some other considerations.