Plane parts:
1. Kill the three bosses (shark, eel, squid) to learn how to make the plane parts... that makes no sense but whatever. DO NOT make the parts until you are ready to use them. There is a glitch where they will disappear after being left alone for some time, and you cannot make them again as far as I know.
2. At the aircraft carrier, create the plane parts and install them.
Fuel:
1. Scavenge two jerry-cans from boat wrecks you come across, as well as eight potatoes from various islands. Making a custom island will make potato harvesting much easier.
2. Create a fuel still. When it is ignited, insert four potatoes into it (the potatoes can be spoiled; they don't have to be edible). This will eventually fill the fuel still with gasoline, which is enough to fill one jerry-can. Do this step twice.
3. At the aircraft carrier, fill the plane's tank with the gasoline.
Water:
1. Create a water still, as well as two clay water jars. Fill them with water to maximum from a water still - they can hold five water drinks each, so you'll need ten fills from the water still total.
2. At the aircraft carrier next to the escape plane, fill the water supply with the water from the jars.
Food:
1. Using a refined knife, go get twelve meat pieces from animals. I found that crabs and fish are the easiest way to get it. The meat can be raw... it doesn't have to be cooked.
2. At the aircraft carrier next to the escape plane, fill the food supply with the meat.
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I recommend using wooden crates to bring all this stuff with you to help with inventory management. Also make sure you are do not have Dehydration or Starvation status on your wristwatch. You can't use the escape plan otherwise.
I also recommend making a shelter-tent at the sandbar in front of the aircraft carrier, as you can save there and re-load from that point if you mess up somehow.