At the expense of listening to hugh whine once more, I've experienced a number of fires during ground operations and in flight, including engine, cabin, cockpit, wing, etc. I've flown through a lot of them, too, as part of regular employment.
In fact, if you claim to have any time behind radial engines and haven't ever had a fire, you've either spent the last few thousand hours deaf and blind, or you lied.
If you have a fire, don't analyze the fire. Don't try to guess at the cause or the circumstance. You need to have a general understanding of what's going on, but don't start guessing weather the carburetor, fuel flow divider, a fuel line, cracked cylinder, broken exhaust, etc, is the problem. Your only concern is making the fire go away.
The biggest danger from a fire is you. I've worked fires on the ground, instructures, in campers, in cars, trucks, homes. I've worked them in fields, on mountain sides, from the ground, and from the air. The most dangerous fire in the world, and the one that kills more people than any, is a grass fire...the one that would least impress you or cause you to think "danger." Which is precisely why it kills people...they don't take it seriously. It moves quickly, behaves unpredictably, and puts out more heat than people expect.
The biggest danger is you...how you react and what you do about it. People think fire, and people think panic. Almost as though the two go together, and that's not a good thing. A better line of thought is when you think fire, think soloution. Fast hands kill. I've had several folks attempt to shut down the wrong engine, also not a good thing. These folks felt that the fire was panic time, and that justified doing things very quickly...failing to realize that hands move faster than thought...which can make things worse, not better. A engine fire or cockpit fire isn't a good thing...but an engine fire or cockpit fire with an exta engine shut down isn't good either. It's worse.
Likewise, I can't count the number of flight instructors I've met who advocate making a beeline for the ground and executing a forced landing at the first sign of smoke. After all, ask any pilot what they were taught as the first justification for making an emergency descent and forced landing, and most will tell you it's a fire. No idea what they'll do once they reach the ground, assuming they survive the landing...and aren't trapped in the wreckage of an aircraft that's already burning. No thought for the fact that the fire rescue that might have been able to help them had they made a beeline for an airport, instead, is no longer nearby or able to help. No thought to the fact that ambulance services aren't readily available like they might be when landing on or near an airport.
Several priorities should become uppermost in your mind when experiencing any kind of fire in flight. The first should be flying the airplane and keeping control...no matter what is going on. That will always be your first priority. I've seen a lot of folks forget that over some relatively minor distractions in the cockpit, including bees, instruments, etc...a fire will certainly be a distraction, and you need to set it in your mind right now that it's not going to distract you from flying the airplane.
The next priority isn't beating feet to the ground for a dramatic forced landing and it certainly isn't making a radio call to tell everybody else about your drama. This is your immediate problem...not theirs. Your next priority is getting the fire out, or under control, and very simply the fire has three things needed to keep living. You want to kill it, and you want to take away at least one of those things.
The fire needs airflow. You're probably not going to smother that fire unless you have an adequate fire extinguishing system. If it's in the engine and you can hit it with a fire bottle or two, that's dandy. If it's in the cockpit, remember that you need air, too. Just like the fire. Remember that if you're using a halon cockpit extinguisher, that halon in contact with open flame becomes phosgene gas, which is extremely toxic, and very deadly. If you deprive the fire of air, get some for yourself.
Fires eat. As infants their requirements are modest, but as they grow up, they eat a whole lot more. A rule of thumb used in structure fires where plenty of food, or fuel, is available, is that the fire doubles in size every sixty seconds. The same fuel structure isn't available in the airplane, but other, more volatile fuels are, and coupled with the extreme amount of air flowing to the fire (compared to a structure or wildfire, an engine fire is a forced air fire...it can burn hot and fast, and grow a whole lot faster), it can grow quickly. The soloution to a fire with a diet problem is to curb the present diet. Starve it for a while. An engine fire is either fuel or oil. Not a whole lot else out there to burn. Shut off the fuel, if it's white smoke. If it's black, you may have an oil fire going, and that's a whole lot worse than a fuel fire. Oil fires go a little slower, but the fire lasts longer, doesn't go out as easily.
Start by pulling back the throttle or power lever or thust lever. If you're in a multi engine airplane, you're going to be doing this anyway to ensure you don't shut down the wrong engine eventually. But you might just put the fire out right then and there. Decreasing fuel flow can do a number of things, but may stop the fire. Or decrease it. See what happens. If it doesn't, your proceedures and probably going to tell you to shut it down. That's fine, follow your checklist. But do everything with a purpose, deliberately, and not in a big hurry. Remember...when you rush...that's when accidents happen. You may be a bigger threat to the safety of the flight than the fire.
Fuel fires don't last long when the fuel is shut off, but oil fires can go on for some time. Oil fires occur when a cylinder lifts and oil dumps onto a turbocharger in a tight cowl, or a fuel fire associated with a mechanical failure sets off an oil fire. Oil invariably needs to leave the engine under pressure, and the only way to stop that is to stop the engine. If it's a propeller driven airplane, shutting down the engine isn't enough, the propeller needs to be stopped. If you can't feather it, you may waste too much time trying to get the prop stopped, and that isn't good...whatever you elect to do given your circumstance, do it on the way to either an airport, forced landing site, or if you elect to get to the ground directly beneath you (also a forced landing site), do it on the way. Don't waste time trying to hold the nose up to get a prop stopped in the hopes of staving off the oil fire...it won't work.
Fires need air. Fires need food, or fuel. Fires need something to keep them alive, an ignition source. Many fires are fairly self-sustaining once born, but you still need to get rid of the ignition source. That may be as simple as turning off ignition. It may be as simple as pulling back the power...a cracked turbocharger or exhaust that's putting hot gas to something and causing a fire will cool faster with the power pulled back. Electrical fires need electricity for ignition...get rid of the electricity. You may still have a fire, but you don't have the reignition potential if you are able to control the fire. Get rid of the source. You may kill the fire, or you may just prevent it from coming back.
Continuing with more thoughts, due to length...