There are an enormous amount of different foods you can use for your Corydoras. The best is of course to give them live food (i.e. mosquito larvae, newly hatched brine shrimp, and micro and grindal worms).
The second best is frozen food such as adult brine shrimp, cyclops, and red mosquito larvae. They are all kind of expensive (unless you grow you own worms such as grindal and micro) and most people I guess use commercial prepared food such as flakes and pellets.
You can also make your own food such as beefheart and shrimp mix. I will expand on that at a later time.

Paul Shuman has written an article about feeding your fish: Cory Corner Housing and feeding, (Aqualink Column)
The Krib, almost all you need to know about food
The Krib, almost all you need to know about live food

Water change

Boring but necessary!
Clear water is not the same as good fish water. The eaten (and uneaten) food you put in the tank is converted to different "poisons" in the tank. To keep the levels down you need to change water regularly. Some of the more sensitive species might need half the volume to be changed every week, while others only need 10% every week.

Diseases

If kept in an aquarium where you regularly change the water, have good filtration, and feed them well, most Corydoras are pretty hardy.

Cory Corner Problems and Diseases in Corydoras, by Paul Schuman (Aqualink Column)
P.I.L.L. Aquarium Diseases