The simplest method of preserving wine is to replace the cork in the bottle, or screw cap where applicable, and store the bottle in the fridge.  This method can be used for both white and red wine.  Keeping leftovers in the fridge will preserve the wine for longer than at room temperature.  When serving previously opened red wine that has been stored in the fridge, just remove the wine from the fridge in advance to enable the wine to reach the required temperature.  White wine can normally be served directly from the fridge.  In general, white wine will keep for longer than red wine when using this method, although both should keep in reasonable condition for up to 3 days at best.

Another method is to invest in a vacuum pump and stopper.  The stopper has a one-way air valve that enables the extraction of air, thus leaving a vacuum in the bottle.  The vacuum pump is used to manually extract air from the bottle.  It is air that causes wine to spoil, and removing the air slows down the deterioration process.
Once sealed and vacuumed, the bottle should be placed in the fridge enabling it to keep for up to 7 days in reasonable condition.

A final method is to purchase a special device that pumps an inert gas, normally nitrogen, into the bottle thus leaving a blanket of gas on the wine to preserve the contents.  This is by far the most effective, notwithstanding expensive, option of preserving your wine.

The theory to preserving Champagne is almost exactly the opposite to that of preserving wine.  Instead of extracting the air, we actually want to seal it in order to prevent air from escaping, which in turns prevents the dissolved carbon dioxide from leaving the wine.  The correct device for this purpose is a simple Champagne stopper which sits on the bottle and clasps the neck of the bottle whilst sealing the opening thus retaining the pressure.  Again, the bottle should be stored in the fridge ready for serving up to 5 days later.

A better method is to use a pressure pump and valve.  This works along the same principle as the vacuum pump, but in reverse.  The Champagne pressure pump is used to pump air into the bottle at a slightly higher pressure than 1ATM in order to maintain an equilibrium with the dissolved carbon dioxide gas under pressure in the wine.  Increasing the pressure in the bottle will slow down the process of the CO2  release. Once the stopper is removed, the pressure in the bottle will reduce to 1ATM and the CO2 should continue to fizz.