The real key to successful training is to start slowly and to work your way up to an event. Join a gym with a pool and exercise equipment so that you can swim, bike and run in one visit and time yourself as you do. Keep track of your endurance and when you seem to run out of steam. Many who start out training for a triathlon find that they need to either bulk up their calf muscles for the biking and running or improve their upper body for the swimming. Some however find that they need to simply lose weight overall so that they can increase their aerobic activity and endurance. For either of these goals, you need to start small and increase your weight lifting or aerobic activity slowly. If you push yourself too hard when training for a triathlon, you can get discouraged quickly and even hurt yourself. As with any other weight training, you need to challenge yourself but slow and steady is also important. Perhaps one of the most important things in triathlon to consider is pacing - if you can't pace yourself properly between the three legs of the race, you'll find it pretty difficult to finish! The swim should literally be a warm-up to the bike ride and the ride a fairly hard effort leading into the run - where you will give it everything you got. Of course, this depends if you're doing a sprint distance, Olympic distance, or up to the full ironman distance.
If your training for a triathlon seems like slow going, consider aiming for competing in a mini triathlon, or a Sprint competition. These are much less distance in all three events and are a great way for new competitors to get started.