I've done a few sprint triatholons, and this is my recommendation as an amature. I wear a one piece bathing suit, with a regular sports bra underneath. I complete the swim, run barefoot to the transition area (which is usually a bike rack set up, you will keep a bag of your stuff or have your stuff laid out here, next to your bike). I would slip on a pair of running shorts over my wet swimming suit, slip into my shoes, and hop on my bike. Then, after the bike ride, you return to the transition area, put your bike back on the rack, and start your run. You'll want to practice these transitions and find what works for you. My husband rode a bike that had pedals requiring special road biking shoes, so he wore those for the bike portion, then changed shoes again for the running portion. My bike could be ridden in normal tennis shoes, so I did that and it saved me some time in transition, since I didn't have to change my shoes again.
Depending on the weather, you will want to put on either bike shorts or running shorts over your suit. If you feel more comfortable, you may also want to put on a t-shirt of some sort. The thing to keep in mind is you want a quick and smooth transition, and you want to be able to change activities quickly (within 1-2 minutes). They make shoe lace toggles, that you can get at your local running shop or sports store, that hold your laces together with a quick slide, so you don't have to tie your shoes.
If this is your first one, I'd recommend researching what to do and ask around for some tips and advice. Ask the race director or coordinator for some tips or info or ask at your local sport store/running shop. If you can financially afford it, they make triathlon suits, that can basically be worn for all of the events. If not, try my suggestions above. Feel free to contact me if you'd like more information/tips! Good luck!