I have about 10 pieces of de Buyer Mineral, some I've been using for a few years now. These comments are pertinent only to the Mineral line.
First, as several have mentioned, the preparation of the new pan can be a bit confusing. Using potato peels is a cheap way to use a natural abrasive and slightly acidic medium to get off the processing oils used in making the material. You can wash it in lightly soapy, hot water, or do what I've done: put boiling water in the pan over high heat, and add a few squeezes of lemon juice. Swirl around, discard, dry the pan, and then go to the "seasoning" with oil step. The potato peel and/or washing stage ISN'T seasoning. For seasoning the pan: add enough oil to fully cover the bottom cooking surface with maybe 2 mm depth; heat and swirl until you start to see a shimmer above the oil (not smoke--that's too hot), then removed from the heat, swirl the hot oil around, and let sit until cool. Discard the oil and wipe it down with dry paper towels.
Secondly, if you've done a first seasoning of a cast-iron pan before, you can do the same here. Basically, the patina that builds up on a carbon steel pan is the same thing that happens when you use a cast-iron pan--it's just more obvious, since the pan isn't black. If you feel that the pan is getting too much of that stuff built up after several uses, you can use boiling water and lemon juice to deglaze much of what you've already built up.
I have two go-to de Buyer Mineral pans that I use for most every sauté and broiling purpose, with the exception of highly acidic foods. The little button that's in the de Buyer handle hole should be discarded--it serves no purpose. My stove is a gas Viking. I've used the Mineral pans for every level of heat on the burner, and with baking and broiling. The handle discolors over time, and using in the oven gives you even more of the baked-on patina from oil splashes, but I've had no issues with using my Mineral pans under direct broiler flame, even when placed an inch or so from the flame.
You have to be careful about deglazing seasoned pans, whether carbon steel or cast iron, if you're using an acidic liquid. The "patina" that builds up over time is mainly lightly polymerized fatty acids from the oils used in the pan, and that material either loosens from the metal or begins to depolymerize, depending on the acidity of the liquid used. You can deglaze with wine as long as you realize that it'll pull off some things that you've cooked in the pan previously, and if you're very sensitive, you may detect a slight metallic taste from the iron ions.
If you do decide to deglaze a pan with wine, you won't hurt the pan, but you may remove some stuff that was cooked onto the surface from many meals ago, and you'll affect the taste of whatever dish you're making.
For cleaning, in my experience, I'd recommend removing the food you've cooked immediately after cooking, then add hot water to help "deglaze" the pan. Bring it to boiling, and push the big bits off with a wooden utensil. Pour out the water, heat the pan until dry, wipe it down with a cloth, and give it a rub with a little dash of oil (rub the oil in with a paper towel to coat the inner surface). Store and use again.
For the oldest de Buyer Mineral pan I have, I usually just have to wipe the inner surface clean with a rag (or boil a little water in it to loosen the spices or bits that remain), then put a little oil in for storage, rubbed onto the cooking surface (in my case, it sits on top of the stove for the next use).