The best way to start is to do so cautiously. Many people try to do way to much, way to soon, and at an intensity level that is way to high. This greatly increases their chance of injury/burnout, and also won't develop their aerobic system to nearly its capacity, and if you start out at too high of an intensity you won't be burning off as much fat, either.
For swimming, find someone who has a lot of experience with it, for instance the swim coach at your local college. Set up a meeting with them and have them watch or even videotape you in the pool. Then, let the corrections of your swim stroke begin. Swimming is 90% form, and 10% fitness, you don't want to learn to swim the wrong way.
For cycling, go to a reputable bike shop and make sure you talk with someone who has racing and bike sizing experience. You need to make sure the bike you get fits you correctly, otherwise your power output and efficiency will be poor, and your chance for injury goes up. Find out when local group rides are. Not only are these great fitness builders, but you can learn a lot from them as well. Don't let the "more experienced" riders intimidate you. Just sit at the back and learn from what they do. Cyclists tend to have egos, so let them do most of the pace setting for the first couple rides you do with them, and they should be happy to help you with pointers.
For running, make sure you get quality running shoes that are correct for your build, then start slowly.
Don't increase your mileage by more than 10% a week, otherwise your risk for injury once again goes up. Try and do some of your running on soft surfaces, such as trails or a rubberized track. Also, once or so a week do 4-6 by 100 meter pick-ups, called "Strides" These are not all-out, but at about 90% of max speed, and make sure you focus on using controlled, smooth form. Do one longer run a week, making it about 20-25% of your weekly milage.
To begin racing, check out
www.cooltri.com There is a link on the left that says "Calendar" This will take you to a list of races for every state, and has a lot of good info on it. There are quite a few shorter sprint of beginner tri's out there.
Also, try
www.trinewbies.com, www.triathlete.com, and
www.insidetriathlon.com. All have lots of good training, equipment and race info.
And I'm sure you've heard it before, but get a physical before starting any training program.
This is a very basic list of how to get started, but I hope it helps. Have fun and be safe.
