Cast Iron
Enameled Cast Iron
Carbon Steel
Oil with a high smoke point- avocado oil, vegetable oil, grapeseed oil
Some brands have specially coated handles or different handle types that are not oven proof. I recommend avoiding those and using removable rubber handles or pot holders instead. The one exception I have to that rule is my wok, which has it's own instructions at the bottom. If you do choose to buy one of these pans, you'll need to follow the manufacturers instructions.
Cast iron or carbon steel pans, skillets, griddle pans etc.
New or Used:
Pans generally say they are ready to use, but in my experience the finish they come with never lasts long or works as well as re-seasoning the pan yourself from the start. If you've received a used pan, you may want to re-season it just so you have the peace of mind knowing how it's been cared for from the "beginning".
Put the pan in the oven on self clean and let your oven do its thing. If your pan isn't real cast iron or carbon steel, you'll find out during this step because this will simply ruin them. But if you buy a familiar name like Lodge or do some research before buying another brand, all this will do is remove the seasoning down to the bare iron and the pan will be fine.
After the pan has cooled, wash it thoroughly with soap* and water and rinse well.
Pour a small amount of oil into the pan and rub all over the inside and outside of the pan, leaving a very, very thin coat of oil.
Put the pan in a 400 F oven for 30 minutes, turn off the heat and let cool. Repeat 3-4 times (there's no reason you have to do this all in one day). Make sure the pan is shiny, but dry, and not sticky. If it is sticky, you've used too much oil. Wash it with soap and water until the stickiness is gone and season again. Once the pan is cool, wipe it with paper towel to make sure there is no excess oil and put it away.
Regular Use:
Whenever you are using it, put enough oil in the pan to just cover the bottom (or the amount called for in a recipe) and get it hot before adding food. Add food as close to the pan as you can; dropping it from far away will cause it to go through the oil and stick to the pan.
When you are done with the pan, immediately rinse it with hot water or pour some hot water into it to soak while you eat and set it back on the stove or on a trivet. If it is a grill pan, you'll just have to leave it as is to cool since it won't hold water and you don't have handles to hold (unless of course you have a very small one with a handle).
When the pan is cool enough for you to handle easily, wash it with a scrubby brush, scrubby sponge, or a stainless steel scrubber. A lot of chefs recommend a chainmail sponge, but I think they are too harsh for regular cleaning. In the case of a grill pan, you can use a small wire brush to get into the grooves.
Rinse the pan thoroughly, then set it back on the stove over medium heat to dry. Once dry, turn off the heat and assess the surface. If it still looks shiny, just let it cool and put it away. If it looks dry, rub it with a very thin layer of oil, then heat it over medium heat until it's piping hot. As soon as you start to see wisps of smoke, turn off the heat and let the pan cool. Wipe it out to make sure there is no excess oil and put away.
If the coating on your pan somehow gets ruined, just strip the pan and start over, but generally this is a rare occurrence.
A note about eggs:
The best pan to use is carbon steel, but it does take some time to get the pan to a point where eggs won't stick. Season the pan two or three extra times beyond what you initially did before you try to make eggs (this can be after using it for something else, you don't have to just season it).
When you are ready, put oil in the pan and heat it up on medium low. Put your eggs in a bowl. Add a pat of butter to the pan and spread it around to melt. Pour the eggs out of the bowl into the butter and cook as desired. When you go to move the eggs, they should slide out easily. If they stick, you need to season the pan a few more times before trying again. Eventually, the surface should allow the eggs to slide easily around the pan.
Woks:
A wok that has a handle of the same material as the pan doesn't need any special treatment, but a wok that has a different handle type (often wood or plastic) can't be put in the oven and will take a little more effort to re-season.
Scrub the pan with a chainmail sponge (the only time I've ever used one), baking soda and hot water until it is as bare and smooth as you can get it. Rinse, then wash with soap and water and a regular scrubby sponge. Rinse thoroughly.
Put on the stove over medium heat to dry, then let cool. Rub all over with a small amount of oil and heat on the stove top over medium heat until piping hot. When you begin to see wisps of smoke, turn off the heat and let it cool. Repeat 3-4 times. Wipe down with paper towel to make sure there is no excess oil before putting away.
Every time you use your wok, you will want to imitate the "wok hei" flavor (basically a smoky, open flame flavor), the best that a home cook can anyway, that good restaurant food cooked in a wok has. Oil the wok all over the inside and put it over high heat. Let it smoke for several seconds, then turn off the heat. Wipe the excess oil out with a paper towel or dump into a bowl to discard. Prepare everything you need, then oil the wok again before cooking your meal.
After the food has been served, rinse the wok or fill it with hot water to soak while you eat. When cooled enough to handle, wash it with a scrub brush or scrubby sponge, soap and hot water. If you ever notice flaking, switch to a stainless steel scrubber and get as much of it off as you can. Rinse throughly and put on the stove over medium heat to dry.
After cooled, assess the finish. If it looks dry, rub a small amount of oil over the inside, heat on medium heat until you just start to see wisps of smoke, then turn off the heat to let it cool. If you had to scrub off flaking, repeat this a few times until the surface is smooth again. Wipe it out to make sure there is no excess oil and put away.
If somehow the finish gets ruined, start over and re-season, but again, this should be very rare.
Enameled Cast iron pans, dutch ovens, and casserole dishes:
Never heat an enameled cast iron pan of any kind empty and don't use very high heat. High heat cooking should be done in regular cast iron or carbon steel.
After using, you can soak the pan if needed. Scrub clean with soap and a scrubby sponge or brush. Do not use a stainless steel scrubber or chain mail. You can even put the pan through the dishwasher as long as there are not a lot of chips.
After the pan has been washed and dried, put a small amount of oil on a paper towel and wipe it over any exposed metal bits, including chips before you put it away. If the color has dulled, which it eventually will, put a tiny amount of oil on a paper towel and rub it all over the outside until it looks shiny, but is dry to the touch.
After awhile, the pan will begin to look a bit grungy on the outside. There are all sorts of tips out there for getting rid of it, but the truth is that as long as the pan is clean, it doesn't actually matter. It just shows that you actually cook with them. If it really bothers you, I recommend using a scrubby sponge with hot water and baking soda. When you've gotten the worst of it, rinse, then wash again with dish soap. It still isn't likely to be pristine, but should bother you less.
If the cook surface of your pan develops large chips, you can technically continue to use the pan safely. However, food is more likely to stick to that spot and it will probably get bigger over time. You will probably want to replace the pan when you can.
*People still often say not to use soap. This is a holdover from when everyone used lye soap on everything. The lye would ruin the seasoning on cast iron. Lye is still a component of many hand and body soaps, but not dish soap. Not unless you are making it yourself. So it is perfectly fine to use and will make cleaning your pans easier.