Differs from state to state – in California, for example, bartending school is certainly an option. Some bars couldn’t care less if you’ve attended one – other, more upscale places might feel better seeing 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, and eventually be brought in to tend bar on slower nights until you get the hang of it. 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. if you want to skip over much of the learning curve. But generally a bar will have a bit of patience with you. Again, if someone accidentally gets a mint leaf in their rum and coke, it’s not going to end any lives.