You can make primitive wicks out of anything that will burn. I saw a video where a woman twisted a part of a sheet of paper into what is called a "SPILL"...just a tight twist of paper used to light pipes and fires in the 19th century.(Why waste expensive paper?) Then it was inserted into the center of a can of vegetable shortening, the top rubbed with a bit of the fat, and then burned. It worked pretty well, and you can store a few cans of the stuff for use with all your other food supplies to get you through till the first time you butcher your animals for food and tallow.
In the long run, though, you need a better and renewable supply of wick making materials. Here is the basic recipe for wicks as seen on You Tube.
Use kite string or any natural flammable string. Cotton is great.
One of the reasons old candles were a problem is that the plain wicks curled in such a way that the wick landed on the candle's melt rim and allowed the wax to run out of the side of the candle. This can be partially solved by braiding three strands of string or whatever, tightly and make the wick out of that. It will curl tightly as it burns and be less of a problem.
1 1/2 cups of warm water.
2 Tablespoons of salt and
4 Tablespoons of borax
Dip your string in the solution for 15 minutes or so.
Hang to dry for several days...I mean DRY!
Dip the wicks in tallow or wax to coat and soak the wick.
Dry thoroughly and store wrapped in paper.
An alternative to this recipe is to soak your wick material in Saltpeter. Saltpeter is Potassium Nitrate. You will see crystals of this on the ceilings of caves, where water has been coming in for some time. There are other ways of making it and they are described here in a Wikipedia article.
See: http://en.wikipedia.org/wiki/Potassium_nitrate It is used also for fertilizers and for explosives. Also, it can be used in meat preservation as in Corning beef.
To make Tallow, you simply accumulate a large quantity of fat from any animal or cooking you have. Any meat fat will do. Put the fat in a large pot and pour in water to cover. Add half a cup of salt to the water. More or less will do depending on how much fat you are processing.
Bring the pot to a boil. Lower the temperature and allow to simmer slowly...about 10 minutes to the pound of fat(this works better if the fat has been ground up or finely chopped.)
When all the fat has been rendered off the solids of the meat by products, strain the mess through a piece of cheesecloth or T shirt material.
Allow the mixture to cool. The fat will rise to the top. Allow to cool completely and refrigerate. When the fat is solid, lift it off the surface of the water and store till ready to use. Dry and scrape off impurities.
You might try this on an outdoor fire or barbecue rather than deal with the smell inside.
Use your new wicks to dip into melted tallow or wax(wax is a luxury that you can only count on if you are keeping bees), with the top of the wick tied to a stick and the bottom tied to a washer to weigh it down. Allow to cool or dip in ice water. Dip again and again till the candle is thick enough. Cool and store in oiled paper in a sealed container in a cool place till ready to use. Remember that these are quite edible and animals will seek them out. These produce a poor but usable candle.
You can also make candles out of sheets of beeswax rolled tightly around a wick.
We used to make an attractive lantern by pouring wax(paraffin) into a large paper milk box. Then we would cool it till a hard shell formed on the inside of the paper. We would cut a square out of the top surface of the wax, then pour out the molten center of the column. Then we would peel off the paper, dip cards or pictures or wrapping paper into hot wax and apply them to the outside of the column. Put a small candle inside the lantern and the light shines through. You can whip paraffin with a hand egg beater to a froth that looks like snow, and apply it to the candle to cover the seams of the pictures or just to decorate. Just apply it warm with a fork.
The colonists used to use fat woods for lighting. Fat woods include spruce and pines as well as any wood with a high oil content that burn easily. A thin sliver of this wood, several inches long was held in a sort of spring-less alligator clip made of iron, usually on a wrought iron stand. The clip was built like a pair of scissors. One of the handles of the scissors was the upright stand, and the other had a weight on it to hold the bottom jaw of the clip closed by gravity.
The wood was put into the holder at a steep angle, near vertical, and simply lit. It provided a dim, yellowish and smoky light.
The knots of soft woods like spruce and pine have tons of pitch in them. There are a couple of ways to exploit this. Sometimes when splitting a soft wood log, the short branches and the knot going into the log will come free. Or, you can cut them out. These can be stuck into the hole in a piece of masonry, stood up between a couple of stones etc. and simply burned.
If you cut a branch very close to the tree, you can split the end of the branch into four parts at right angles. Pry the splits apart slightly and push in some shavings of wood or paper. Find a secure place to stand the torch and light the split end. These drop plenty of pitch, usually burning and hard to put out, so be careful where you put these. Allow them to drip over masonry, ceramic, metal or glass.
If you can find a source of rushes, the outer covering of the stem can be peeled off and the pithy core can be soaked in oil or fat and burned like a splint in a holder. I have heard that cat tails(including the big brown head) can be used, but I have never seen them.
Oil lamps have been in use for thousands and thousands of years. Possibly cavemen used shells with meat oils and fats in them to make light with rush wicks. They are easy, though not super bright. They were often made of ceramic, the rich having metal ones in bronze and iron on beautiful stands, and in more recent centuries most were of iron. They are shaped a bit like a gravy boat. Sometimes the pour spout was isolated from the main reservoir like a short spouted teapot, or the thing could be just a reservoir with a spout simply swelling out from one side. A wick, rush or a scrap of cloth is laid in the oil or fat, inside the spout and this wicks up into the cloth to feed the flame. The hard fat is sometimes hard to get going unless you have soaked the cloth wick in fat first. Oils may work well but are harder to come by in North America.. Watch out for genies!
Thank you Wikipedia for the use of this picture of a rush light. This could be used with fat wood splints or rushes. These are fairly common to purchase on line as reproductions or even antiques.