To begin your umpire career, start when there is snow on the ground. Most associations have their rules and mechanics clinics well before the season begins. The first thing to do is to contact the Umpire in Charge, or UIC, of your local youth league. An internet search of Little League, USSSA, Dixie Youth, Babe Ruth / Cal Ripken, and AAU in your area is a good starting point, as is calling your local parks and recreation department. The earlier you contact the UIC, the better. Ask for information regarding uniform shirt color(s), hats, pants color, fees, and the date and time of any rules/mechanics meetings. Additionally, many leagues will pay their umpires' annual fee for the national association membership, so ask about it.
Attend as many of these rules/mechanics meetings as you can with as many leagues as you can. I recommend joining as many leagues and associations as possible. Even if you end up declining games, you will have more opportunities to work and will receive a broader initial training.
Read the rulebook. The Rules of Major League Baseball is a great place to start. I also highly recommend studying the PBUC Umpire Manual. This manual contains interpretations, clarifications, general practices, and rulings endorsed by Minor League Baseball and the Professional Baseball Umpire Corp. It is provided as a supplement to the Official Baseball Playing Rules which govern all games in Minor League Baseball.
In addition to a basic understanding of the rules, you must have a specific knowledge of the rules of your association. Get a rulebook for your association. Fortunately, most rulebooks are available online for free download.
If you have any ambition in continuing into higher levels, you may want to join a local umpire's association and high school sports association.