As with any other sport you might play, except perhaps a child's game of tag, you're going to need some equipment for a triathlon. If you're serious about competing and being successful in your triathlons, then you'll need proper triathlon training gear. The right gear can help you succeed whereas the wrong triathlon training gear will do nothing but waste your money.
You might not think of your wetsuit as part of your triathlon training gear but it really is. The right suit is going to help keep you insulated in the water, as even on the warmest of summer days water can still be very cold. It can also help keep you buoyant, and triathlon suits are built to be worn through all legs of the competition so that you don't need to waste time changing between events. The reason you want to consider your suit as part of your triathlon training gear is that some people are a bit uncomfortable with a suit when they first start out, being used to a regular swimsuit in the water and bike shorts while riding. Suits are designed to be very comfortable and lightweight, but you should still practice all three parts of your competition in one and consider it as being part of your usual triathlon training gear so that you know exactly what to expect on the day of your competition.
The right triathlon training gear for your bike is also important. So-called aerobars, which lean you forward while riding, allow the largest muscles in your legs to work the hardest which then cuts down on fatigue and can give you greater control of your bike over rocky terrain. Racing wheels are also considered by many to be invaluable triathlon training gear equipment especially if your biking part of the competition is held indoors. These wheels can really help to shave some valuable minutes off your overall speed and time.
Of course triathlon training gear should include whatever is right for you. If you need to improve your running time you may need to include weight lifting in your routine to build up your leg muscles; the same for swimming, which requires upper body strength. It's important to be honest and realistic about the parts of a triathlon that you need to improve upon so that you can improve your training routine for yourself personally.