First you need the proper equipment. There are people who open bottles of wine everyday for a living. These people are called waiters. There is a tool called the waiter's key. This is not a coincidence. Proper ones look like this:
The thing on the left is a foil cutter. You can get by without one, but they're very cheap and make your life a bit easier. If you don't have one, you must use the little "knife" on the back of the waiter's key.
The important thing is to get a waiter's key that has the hinge. To do it properly you have to have the two stages. See the orange arrow.
Use the foil cutter to remove the foil. One twist while pressing inward and it should come right off cleanly.
Next open up the waiter's key and insert the corkscrew. You do not want to push the tip directly into the center. Instead, you want to do it slightly off-center so that the center of the corkscrew spiral is aligned with the center of the cork itself.
The next part is the part that so many wine-opening gadgets get wrong. You want to twist the corkscrew down far enough into the cork so that you can maximize the volume of cork you exert force on while pulling it out. However it is of utmost importance that you
do not puncture the bottom of the cork. Things like The Rabbit and counter-mounted cork extracting machines almost universally puncture the bottom of the cork. Those weird cork extractors that have the two arms that go up while you screw the thing downward are also notorious for doing this--they can't function properly without breaking the bottom of the cork. If you break the bottom of the cork, depending on the age and composition of the cork, you most likely end up with bits of cork floating in the wine. Which nobody likes.
With the waiter's key you generally want to leave about a full twist above the top of the cork. Once you are at the proper depth, put the first stage of the metal arm on the rim of the bottle and make sure the hinge is bent inward.
Then you simply left the lever while holding the metal arm in place with your free hand.
When the cork cannot be pulled further, move the hinge so that it is bent outward and put the second stage of the metal arm on the rim.
Pull upward on the lever one more time and the cork should slide right out with no pulling or problems.
At this point you remove the cork from the cork screw, make sure the bottom of the cork is wet by touching it with your finger and smell the wine in the bottle. If the cork is dry and the wine smells like socks and nasty, find another bottle. Otherwise, drink and enjoy.
If at any point the cork doesn't move easily, jiggle the lever up and down or give the screw another bit of a twist and try again. Basically approach it as you would any stuck item by trying to loosen it by applying forces in different places until it comes unstuck.
