For most tris, people wear a tri top and shorts or one piece tri suit under their wetsuit. You would want something tight-fitting and not too bulky as you do get wet under a wetsuit. Regular running clothes would probably be too loose; regular bike shorts will really absorb a lot of water in the chamois
(pad
). In theory, it's possible to just wear a swimsuit under the wetsuit and then put on a top and shorts over that after the swim; in reality, it's difficult to get stuff on wet, you would probably chafe with bike shorts over a wet swimsuit, and you can't get nude in transition to take the swimsuit off
(usually only full ironman and some, but not all, half-ironman races have changing tents
). For full IM, some people do wear a swimsuit for the swim and then change into a bike or tri top and shorts to be more comfortable, but that's only possible with change tents. You won't find those at a typical sprint or Olympic distance.
Really your best bet is a tri suit or tri top
(or a tight-fitting run camisole
) and shorts. If female, you could just wear a sports bra on top for the swim and then pull on a jersey or tech shirt in transition for the bike and run legs.
(I think IM and USAT rules require torsos for both genders to be covered for bike and swim--I could be wrong about that as until this year I've mostly raced in Asia where rules can be different.
) Compression shorts would be okay on the swim but are not the most comfortable for the bike. They might be okay for a sprint. I did my first tri
(non-wetsuit swim
) in a sports bra and compression shorts, with a tech T-shirt for the bike and swim. Not the most comfortable getup
(I had some chafing on the bike
), but it served.