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.

The best foods to make is protein swimming in lots of fat. For example, shallow-fried frikadellen. Take care to not simply use the pan as a griddle and spread a tiny layer of fat, fry in at least half a centimeter of fat, if not more. Meat and (not scrambled) eggs are good, but don't use fish.

Take care to have pure meat and not any short carbs (sugar) or long dry carbs kicking around. For example, bacon strips from bacon brined in a sweetened solution are bad. This stuff will caramelize on the pan and make it stickier than before. Bread or very doughy griddle-cake varieties (english muffins, etc.) are also bad, as they will char onto your pan.

Some doughy things are good for the seasoning, if they are not stick-prone. I find that Ruhlmann style crepes work great (2 parts egg, 2 parts milk, 1 part flour, by weight). Make sure to use enough oil before the first crepe. For later crepes, watch the pan permanently, and turn the crepe in the short time after it has self-released but before it has burnt onto the pan. You might need to play with the heat, remember that cast iron reacts slowly. Mekiza/lagos will also work well, if the pan is deep enough that you can comfortably fill it with enough oil (it doesn't have to be enough for deep frying, but you still need over a centimeter, and then enough space above it to avoid large splashes).

I wouldn't do vegetables at first, because they tend to sweat a lot of water, and may char onto the surface if you leave them alone for a short time.

Also, don't use butter at the beginning. Lard works really great for the beginning in a seasoned pan, coconut oil is also good. If you are using less saturated oils, don't do lineseed, that's very sticky when it polymerizes.

Once the seasoning "holds" better, you can expand to other things, of course. Still, your grilled cheese sandwiches are not something I would do in a seasoned pan at all, even with established seasoning.

When I get a new cast iron (pre seasoned or not) I always add a thin layer of oil to the entire skillet, usually vegetable oil, and bake it in the oven at 325° for about an hour. I also do this every now and then between uses to make sure it remains well seasoned and non stick. Also, to help prevent sticking, preheat your pan well before cooking. Hope this helps. Good luck!

I love to deep fry stuff, even like french fries. Most of the french/european carbon steel/cast iron pans like you to fry potato skins inside the pan before using. Well after you have thoroughly cleaned the cooking surface. Peel the skins off the potatoes and then fry them in the pan. then when the skins are nice and burnt and crispy throw them away. Clean the pan and then deep fry the left over potatoes as either chips or as fries.