First of all, if you want to avoid the stereotypical "bad tequila experience," be sure to look for "100% Agave" on the label. The reason Cuervo Gold and other shitty "gold" tequilas cause such nasty hangovers is that they're only partly made from agave (the plant from which tequila and mezcal are made). The rest is from sugar or lord only knows what else, and it's that part that makes you sick.
Once you've narrowed the field to 100% agave tequilas, there are three main types: blanco (or silver), reposado and anejo. The difference is how long they have been aged, anejo being the longest. Blanco makes a great margarita, although some people also like sipping it. Reposado can go either way--sip it or mix it. But anejo should not be wasted in a mixed drink--it should be savored like a fine cognac.
If you do decide to sip a good tequila, try a chaser of "sangrita." There are many different recipes (you can find plenty with a Google search), but it's generally like a spicy bloody mary mix. It's better than the "lick salt, take shot, suck lime" ritual (although there's a time and a place for that too, don't get me wrong!).
Finally, a word about mezcal. Mezcal is actually a generic term for alcohol distilled from the agave plant. Tequila refers specifically to the mezcal made in the state of Jalisco (sort of like a French A.O.C.). What is sold as "mezcal" often comes from the state of Oaxaca, and it can be quite enjoyable too. It's less "polished" than tequila (my wife has likened it to "licking an old tire"), but I actually like the smokiness of it. Yes, some mezcal has a worm in the bottle (note that tequila NEVER does), but the better brands don't.
So, as with any type of alcohol, there's good stuff and bad stuff. Just as it's not fair to generalize about wine after tasting the stuff from a box or a jug, it's too bad that people get a bad impression from one rough night of Cuervo Gold in college. Unfortunately, this seems inevitable--unlike wine, where the good stuff is well known and widely available, most people just don't know any better when it comes to tequila. Luckily, good tequila is gradually becoming more popular, so there are now more and more restaurants and bars that stock the better quality brands. Give it a try!
Common sense is not so common. -Voltaire