I don't know what most people do, but I generally wear the same clothing for the entire race. Either shorts/singlet combo or a one piece suit. Or maybe just shorts if wearing a top is not required.

For swim-to-bike, I take off the goggles and swim cap, put on sunglasses and helmet and cycling shoes and hop on the bike and go. For bike-to-run, I take off the helmet and cycling shoes, put on running shoes and go. I don't wear socks but put a little Body-Glide on my feet before the swim to prevent blisters.