If you're looking for the "nuts & bolts" of it, it's actually not that difficult. Basically, you find out if there's any interest in having a union. You're allowed to do this at work, as long as you're on a break and not "on the floor" so to speak. You pretty much have to be away from anyplace where people are actually working, so a break room or cafeteria usually work best.
If there's interest, you'll need to start collecting signature cards (a "card check"). This is pretty much just a "sign up" list of people who want the union. If at least 30% of the employees sign up, then the National Labor Relations Board comes in and holds an election (remember the vote scene in "Norma Rae"? Same thing).
If over 50% (a simple majority, meaning half plus one) sign up, then the vote is skipped and you have your union.