What foods to cook in a new cast iron skillet that will help improve the seasoning?
To improve seasoning, any frying or searing will help. Frying chicken, searing steak, etc. Anything that has to do with heating oil. The things that can damage the seasoning are acidic mixes, and in long duration. An example: Simmering tomato sauce.

I would also like to add that actually seasoning the pan, and proper handling and storage will give you much better results in improving your seasoning. One good starting point is Kenji Lopez alt's advice on cast iron pans. There are other very good sources on this too.