The single most important thing you need to pay attention to when choosing the right wine glasses is the wine glass shape. The wrong shape in a wine glass will completely alter the wine tasting experience, even to the point where you can't properly examine or smell the wine in order to evaluate it. You don't necessarily need to get a different glass for every type of wine - though that is possible - but you should try to have a different glass shape for white wine, red wine, and sparkling wine.
Why Do Wine Glass Shapes Matter?
Wine glass shape affects how the wine tastes because the shape of the wine glass affects how well the wine aerates. A narrow glass shape allows very little surface area for the wine, preventing it from aerating well. Narrow glasses also force you to fill them up fuller, preventing you from swirling the wine to increase its surface area. Finally, narrow glasses don't have enough room for you to put your nose into the glass and breathe in deeply, which is necessary to best experience a wine's aroma.
Wine Glass Shapes: Red Wines
As a general rule, red wines require glasses that have the widest bowls. Because red wines are made with the grape skins, they can have far more complicated compounds in them than white wines which are made without the skins. It is easier to appreciate all of a red wine's complex aromatic compounds with a glass that makes it easier for you to entice those compounds out of the wine.
With a larger bowl, you can pour more wine into the glass while still being able to swirl around quite a bit. It's important, though, that the top of the glass become narrow rather than remain broad. A narrow rim ensures that the air in the glass, now filled with the wine's aromatic compounds, doesn't quickly escape. This allows you to put your nose in the glass and breathe in the mixture.
Wine Glass Shapes: White Wines
Because white wines are made without the skins, glasses for white wines don't need bowls that are quite so broad nor do they need to narrow by as much at the rim. You don't want perfectly vertical sides to the glass, but you will do just fine with a glass that is simply a smaller version of what you use for red wines. Because it can be an imposition to have very many good wine glasses, you can usually get away with just one sort of glass for both red and white wines. It doesn't need to be as large as a traditional red wine glass, but it's probably more important that it not be too small.
Wine Glass Shapes: Champagne & Sparkling Wines
Although you can get away with a single glass for both red and white wines, you shouldn't use the same glass shape for sparkling wines. Not only is there no need to breathe in as deeply the aromas of sparkling wine, you definitely shouldn't swirl it around in your glass. For this reason, sparkling wines should fill up their glass, which should be a champagne trumpet, tulip, or flute. These champagne glasses are named for the shape they resemble: the trumpet looks like the bell-end of a trumpet, the tulip looks like the head of the tulip flower, and the flute is simply a long, narrow glass like a flute.