Do I bike with my wet bathing suit on???

You should post this in "Introduce Yourself" or "Triathlon Talk" Forums up above. Some other information you should include in your post:

Is this a Pool swim or Open Water Swim (lake)
What is the goal for your first...To Finish (time not an issue) or Have a time in mind (want to compete)
Do you have the funds to purchase other Tri Related race gear (tri shorts and top; tri suit) that can act as swim suit and bike clothing or are you on a limited budget

In general, yes, you bike and run while wearing the same thing you wore in the pool. Most people invest in triathlon shorts and a triathlon top. These are designed to be quick drying and the shorts have a thin pad designed to not hold too much water and not chafe when wet.

If this is an indoor triathlon you may have time to pull regular biking shorts on over your swim suit, especially if they give you a set transition time.

Apologies I forgot to answer your original question...

In most case you wear on the bike what you wear in the water if it is a pool swim; open water if you are wearing a Wetsuit you would remove it.  If you wanted to wear bike shorts you could put those on in Transition and take them off for the run and switch into a pair of running shorts or wear your bathing suit.  Time it takes in Transition to put on a pair of bike shorts when you are wet might eat up some time there go the reason for investing in a trisuit or tri top and tri shorts.

You can invest in trishorts, which I highly recommend unless you are on a really tight budget.

But you MUST invest in some sort of lubrication for your tush.  I like Bodyglide, but there are plenty of other products.  Put it right on your skin whereever you think that you might have friction.

If you have not already found the BC group, head over to

http://www.beginnertriathlete.com/discussion/forums/forum-view.asp?fid=92#s

there are folks who have done this tri before and will offer advice -  if I recall correctly, it has been / can be quite cold....

Good Luck!!