People ask which of these is best. None of them is. They behave differently enough that a kitchen with all three is not over-equipped, and a kitchen with the wrong one for the job will fight it every week. Here is what actually separates them.
Tri-ply stainless
Three bonded layers (stainless on both faces, aluminium in the middle. The aluminium spreads heat sideways so the pan does not have a hot ring where the burner is. The stainless faces mean nothing reacts: tomato, wine, lemon and vinegar are all fine for as long as you like.
It is the only one of the three that needs no maintenance ritual. Wash it, dry it or don’t, put it away. It goes in the oven, it takes metal tools, and there is no coating to wear out, so a good tri-ply pan is a twenty-year object rather than a three-year one.
The trade-off is that it sticks until you learn it. Stainless holds food until a crust forms and releases on its own, which means preheating properly and then waiting instead of poking. It is also the most expensive of the three per pan, because bonding three layers of metal is more work than casting one.
Cast iron
One piece of metal, usually 4 to 5 mm thick, and heavy) a 26 cm skillet is around 2.4 kg. That mass is the entire argument for it. When a cold steak hits a cast iron pan the surface temperature barely drops, so you get a crust instead of grey water. Nothing else in a domestic kitchen does that as well.
It is slow. It takes several minutes to come up to temperature and it does not respond quickly when you turn the heat down, so it is a poor choice for anything that needs fine control. Bare cast iron needs seasoning and it rusts if stored damp. Long acidic simmers strip the seasoning back.
Enamelled cast iron removes the seasoning problem and the rust problem, and adds a new one: the enamel chips, and a chip cannot be fixed. It also cannot go under a grill.
Carbon steel
Chemically close to cast iron but rolled thin, 1.5 to 2 mm, instead of cast thick. That makes it much lighter and much more responsive. A carbon steel wok or frying pan changes temperature within seconds of you changing the flame, which is why professional kitchens use it and why it suits stir-frying.
It seasons like cast iron and rusts like cast iron, so it carries the same maintenance. It holds less heat than cast iron, so it is worse for a thick steak and better for anything you need to move quickly.
If you are buying one pan
Buy tri-ply. It is the most forgiving to own even though it is the least forgiving to cook in, and it will not deteriorate while you work out what else you need. If you are buying two, add cast iron for searing. Add carbon steel third, when you already know you enjoy the maintenance.