I did my first season of triathlon last year and wore a ShockAbsorber sports bra under a tri-suit. Yes, it gets wet, but with tight fitting Lycra it doesn't feel horrible and soggy. You really don't notice you are wet and once you get on the bike you pretty much dry out quickly. I would advise either a tri-suit or tight fitting Lycra separates as baggy cotton would be horrible.
Yep- my norms are fairly big and I wore a run bra and tri suit under a wet suit. You won't notice as being damp is part of the experience - I agree with thistle - wear tightish Lycra and avoid flapping t-shirts. I'm no skinnie-Minnie and it felt fine.
It's amazing what you get used to wearing. My friend commented that having to wear a Lycra tri suit was why the training was so hard. The event photos take no prisoners tho- I look like a hippo coming out of the sea last year but will now always remember to take my swim hat off before running up the beach - not a pretty sight!!