Bartending qualifications differ from state to state, but for the most part it's a question of being of age and of willingness to pour booze into drinkware. In some places, bartending school is almost a requirement to get a job. Then there are some bars that couldn't care less if you've attended one. The more upscale places might feel better about hiring you if they see a certificate of some kind.
For the most part, however, bartenders learn as they go. Many start out as servers, so you might cut your teeth at a Denny's, then advance to a restaurant that serves liquor, do some cocktailing, and learn the bar that way.
There are plenty of books you can check out on your own to learn the bizarre names of drinks, how to make each variety of cocktail, etc. But generally, a bar will have a bit of patience with you. If someone accidentally gets a mint leaf in their rum and coke, it's not going to end any lives—it might even add some flavor.