Swimming needs are basic. You will need swim suit, and a pair of goggles. Goggles are not made to fit all face types so you may need to try a few different types to find the ones you like the best. A good rule is to push them on over your eyes without the strap over your head and see if they stay in place. Well fitting goggles should stay in place for a few seconds.
You are going to need a bike. But don't run out a purchase one. For the first race or two you can use a bike you already own or borrow one. Many people I know have completed their first races on a mountain bike. This will give you time to learn about what type of bike you want. When you finally buy a bike or upgrade to a road or triathlon bike you want to take time to find one that you feel comfortable riding. You will also need a bike helmet to protect yourself in case of a crash. Finally, you will want some sunglasses to wear while riding. They help you to look good but they also keep bugs and other small debris from getting in your eyes. There are other biking clothes you can purchase but they are not necessary for getting started.
The most important running equipment is a good pair of running shoes. Make sure you go to a running store for your first pair and have them help you find a pair that matches your running stride. Good running shoes will make all the difference. I also recommend purchasing some running clothes that wick sweat away. For your first triathlon you will wear your swimsuit for all events so you don't need to worry about running shorts during the race, but some people like to put shorts on over their suit prior to the bike leg of the race. You can also use your running shirt as a biking shirt and put it on after the swim and wear it for the rest of the race.