Q: What kind of frying pan (non-stick, stainless, cast iron) should I use?

I am not sure that there is a definitive answer to your questions, but here are a few hints:

• Look at the recipe.  If it calls for making a pan gravy or sauce in the pan after meat or other ingredients are cooked, then you want to use a pan that does not have a non-stick surface.  The brown stuff that is left in the bottom of the pan, called the fond, is what makes the flavor base for the gravy.  You want it to be left in the pan and not stuck to the food.  On the other hand, if the pan is not deglazed, more flavor will stay with the food if you use a non-stick surface.

• If the recipe calls for the pan to be placed in a hot oven, as is frequently done with chicken or duck breasts or pork tenderloin, make sure that the pan will withstand the heat.  Most Teflon pans are rated only to 350°F (175°C).  Anodized aluminum may go up to 400°F (205°C).  Plastic and phenol handles, however,  may not be able to tolerate those temperatures or may become brittle from repeated heating.  Stainless steel, un-coated aluminum and cast iron pans with metal handles can stand up to most oven temperatures.  Always check and follow manufacturer's directions.
• Use the heaviest frying pan that you can comfortably work with.  A pan with a thin bottom is more likely to have hot spots where food will burn and is more likely to warp with heat.  Also, a lighter pan will cool down more as food is added, which may affect the final result.
• Always use a pan that will comfortably accomodate the food you are cooking.
• Some cooks claim that Cajun Blackened Fish, Corn Bread,  or other recipes can't be made successfully in anything but a cast iron pan.  This isn't a problem which I have encountered, but if you are a purist, you may need a cast iron pan for such recipes.

Ultimately, the choice of pan depends on a number of factors, including budget and the types of foods you like to cook.  If you cook a wide variety of dishes and have the budget, look to getting several pans to suit your needs.  If not, pick one good quality pan that is the most versatile for your cooking style.