Generally, you register to vote. As long as you input plausible information and a real address you can receive mail at, you are likely to get on the rolls. Once you get on the rolls, you just show up at the polling station and tell them who you are (no ID required), they look for the name in the book, mark you as having voted, and you go vote.
Several years ago they did pass a law federally requiring an ID if you couldn't provide the last-four of an SSN or a driver's license, though, if you're a new voter. Not sure how that works if e.g., you're Amish and don't have an SSN or driver's license.