There are four types of molecules that create pigments in plants: carotenoids, anthocyanins, betalains and clorophyll. With the exception of chorophyll, all of these molecules can contribute to producing a red pigment. And as a complete opposite to the controversy surrounding health affects of artificial red food dye, red pigments in food nearly always indicate very healthy antioxidant substances.
I've divided the red foods based on the solubility of their pigments. Both groups can be useful as accents, or even as pigments to dye other foods. But it's useful to know the difference so that you can know what to expect when you're composing a dish. Since most foods have a large quantity of water in them, water soluble red pigments will easily bleed from one to the other. So that's why the roasted beets in your salad will turn the lettuce red. Oil soluble pigments will not spread as easily (which is why tomatoes in the same salad don't turn the lettuce red). Generally, if you're looking to dye a non-red food red, you'll want to look to a water soluble food, like beets or berries. If you're looking for an individual element to keep its color (perhaps amidst other colorful elements) then look for a food with a fat soluble pigment.
Beets - Perhaps the most brilliant and potent red food. Try using concentrated beet juice in place of red food coloring.
Berries - Raspberries, Cranberries, Currents, Goji, Strawberries are all the most red when they are fresh and in season. Berries make a star feature to a dish whether they are cooked, pureed or just left raw.
Rhubarb - Rhubarb can lose it's red pigment easily during cooking. Try saving any liquid leftover from cooking, reducing it and tossing the cooked fruit with the concentrated, colorful liquid.
Grapes & Stone Fruits - Cherries, Plums, Grapes all have lots of pigment in their skin. Red wine gets all of it's redness from the grape's skin. Try roasting dark colored plums with their skin on. Some of the pigment will leach through to the fruit below. Remove the skins gently after they have cooled and you'll have a gorgeous red fruit.
Pomegranate - Pomegranate is not a berry, but most of the cooking implications are the same. Fresh, it's a beautiful accent for salads and fruit dishes. Pomegranate juice also makes a colorful food dye.
Citrus - When they are in season, red grapefruit blood orange display some striking reds. The color holds up cooked or raw.
Hibiscus - Dried Hibiscus is commonly available as an herbal tea. Infusions of hibiscus have a very tart flavor and are a brilliant ruby red. Try using hibiscus syrup to sweeten lemonade or make stunning mixed drinks.
Red Grains - Rice and quinoa both have heirloom grains that are red. Easy way to add color to a plate!
Sumac- It might look peppery hot, but sumac tastes more smoky and lemony. Try a sprinkle of sumac on roasted fish or mixed into a yogurt sauce.
Tomatoes - In peak season, tomatoes come in all shades of red. As tomatoes cook, the red darkens and becomes more brownish. Fresh tomatoes and peppers make perfect elements to add contrast to a dish.
Peppers - The pepper family is broad in flavor and color. Fresh peppers, roasted peppers and all manner of dried and powdered peppers can provide a brilliant red accent without dying their neighbors red.
Achiote - Achiote (also called annatto) is a powerful pigment as well as a seasoning. Look for little boxes of achiote paste at well stocked grocery stores or stores that specialize in Mexican or South American ingredients. Try heating some cooking oil with a little achiote paste, or some whole annato seeds. Let the oil infuse over low heat for a few minutes, then use your colorful oil as an accent to fish or poultry dishes.
For anyone interested in further reading, I've written a post on my blog about why the color of food seems so darned important.