It's most normal to start out as barback, or busboy. This guy clears glasses, stocks fridges, washes glasses, gets ice, cleans up after the bartenders, wipes down the bar etc etc.
After a while, there is certain to be a position come up where you can step up to the bartenders position.
Also (and this is a BIG rule for me in my restaurant), if you want to move up in the world, LET ME KNOW. I can't mind-read and I won't know that you want to become a bartender if you don't tell me right from the start.
I'm not going to come over to you and say, "Hey, wanna be bartender?"
Put your hand up and let people know what you are aiming for.