Serious business begins...
First of all, store the bottle vertically 3 days before opening, this to prevent any deposit from mixing with the wine at the time of handling.
First cut the capsule and wipe with a damp cloth the top of the neck and the cap.
It is "normal" for an old cork to be moldy.
Then I usually use a bladed corkscrew:
By pushing the blades between the cork and the glass of the neck and turning, first without lifting, you take off the cork from the neck of the bottle and then, while turning, you have to try to gently remove the cork.
Twenty rounds are sometimes necessary.
Tip: to prevent the cork from sinking when inserting this tool (this can happen with corks over 40 years old), you can wedge a knife diagonally in the cork against the neck of the bottle during the insertion of the tool, it will lock the plug.
It is quite common for the corks to break into several pieces and require a shellfish pick to extract the remains.
Once the cork is removed, leave the bottle in a cool room (15/16° for a red wine) or in the refrigerator for a white or sweet wine and serve at least 3 or 4 hours after opening, without decanting, the a sudden supply of oxygen could damage it.
I have planned a blog post on this subject and I will put explanatory photos as well as other possible methods.
The attached photos show the (difficult) opening of a 1982 Châteauneuf du Pape, I had to repeat the maneuver 4 times!!!
Quite average quality cork, but the tasting was great: superb jammy and liquorice notes!!