Propagating palm trees is one of those things—because of the way they grow, palm trees cannot be propagated through the asexual means used to propagate many trees. Air layering, cuttings, and division are for the most part not effective when starting palm trees. Usually, the only way to start a palm tree is from seed.
Palm tree seeds can be obtained either through mail order or from flowering trees.
The seeds of most palms are held on branching fluorescences and vary in appearance depending on the species. Some are small and bright red, like berries, while others like the coconut are more instantly recognizable. It's best to use fresh palm seeds if possible because they tend to sprout more readily. To test if a palm tree seed is viable, drop it into a bowl of warm water. Seeds that float are no good—they lack internal organs called endosperms that are necessary for reproduction. If the seed sinks, it's more likely to be viable (an exception to this are coconuts, which can sprout after floating for a long time).
To actually sprout the seed, plant it in a small container with a very thin layer of soil, or even only half buried. Palms do not readily sprout if they are buried too deep—in nature, palm seeds are dispersed by the wind and animals and are rarely buried before they are expected to sprout.
Once you planted the palm seed, move the container to a very warm, very humid place. If you have a window-box in a steamy bathroom, the location should be perfect. If you don't, wrap the container in a plastic bag or plastic food wrap and place it in a warm location, like the top of your refrigerator or a warm windowsill.
The time to germination varies wildly among palm species, but it's probably longer than you're accustomed to. Some palm trees will sprout in 70 days, others, such as coconut palms, can easily take six months to sprout. Don't worry if the seed starts looking a little ragged while you're waiting. It's not uncommon for palm seeds to shrivel and otherwise look dead before they actually sprout.
Once the plants have sprouted, move them to a very warm place (at least 75˚F) with relatively high humidity. The notable exception here are palms from arid regions, which should be exposed to less humidity as seedlings. Palm seedlings also need abundant light, and many species will thrive in dappled sunlight. If possible, sprout your palms over the winter or in early spring and move the young plants outside for their first summer.
Many popular species of palm are susceptible to root shock from transplanting, so it's not a good idea to transplant young palm trees until they have at least three or four sets of leaves. For this reason, it's a good idea to start palm seedlings in 4" containers at least. The rate of growth for a palm seedling will vary depending on the species. Some palms, like Bismarck, are very slow growers, while others, like the Christmas palm, will grow rapidly after they sprout.
Palm seedlings do not require fertilizer at first, but once they start actively growing, it's a good idea to start them on a weak liquid fertilizer. Use a palm fertilizer that has been formulated especially for palm trees. It's a misconception that palm trees don't need fertilizer—in fact, the different between a good palm and a great palm is fertilizer.
Once the palm has developed several sets of leaves and the leaves begin to assume their mature form, you can move the plant inside and transplant it into a larger container.