By definition, a "desert" island would be arid, lacking fresh water and foliage. A better term would be "deserted" island, which would be an island without human inhabitants. This topic is a favorite of speculation, and so there are many opinions. The most important factor is water; humans cannot survive for very long without fresh drinking water, anywhere from three to five days at maximum. Rainwater and fresh springs are best, but might not be present. To obtain drinking water, you must search for dense foliage, preferably below sea level. Moist dirt and mud can be gathered and strained through fabric, or allowed to drain into a container; it is likely that this water will contain bacteria and other harmful particles, so the water should be sterilized through boiling, if possible. You can die just as easily of dehydration through vomiting or diarrhea as you can from simple thirst. If plastic is available, a solar still can concentrate up to a cup of fresh water each day.
For food, it is first important that you know how to identify poisonous or harmful plants. Many poisonous plants look just like safe plants; it is easy to make mistakes. Never eat mushrooms! They are more likely than any other food to be poisonous. The best food group would be vegetables; safe root vegetables such as potatoes or carrots are easily picked and cultivated. If you can find a natural plot of beans, conserve and cultivate them; beans are one of the most nutrient-dense foods in the world, and soybeans contain the only plant-based complete protein.
If possible, catch fish; their protein is healthy and very renewable considering the size of the ocean. Set out nets or lines, and smoke the fish meat as soon as possible to preserve it from decay. You can collect other sea-based foods as well, such as shellfish (mussels, clams, crabs) and seaweeds, which by themselves are very nutritious. Also, gather any fruits available and save the seeds.
To sum up, first find a source of fresh water, and then worry about food.