The main reasons that our cats become "finicky" are because we get them used to eating only one kind of food, we offer people food and they get hooked on the strong taste, and we feed the so called "gourmet" cat foods that are high on palatability and low on nutrition. The best place to start is by asking the advice of your veterinarian. She can guide you to the correct food based on your cat's age, health and activity level. Once you know which food to feed, start introducing it slowly. Mix a tiny bit into the food your cat is currently eating. Gradually add more of the new food each day. This will help avoid rejection of the food and will also avoid any stomach upsets.
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