When do you carry fluids?

For me, it totally depends on the heat/humidity.
I seldom have to carry water, since there's an abundance of water fountains along the lake here in Chicago. Anything over 15 miles, I'll bring Infinit mix.

As an example, I regularly run home from work, about 6.2 miles, last night, it was 65 and dry, I didn't stop once to drink. Over the summer, when it's 80's and humid, I'll stop 3, maybe 4 times.

I tend to be overprecautious regarding hydration.
For your upcomng race, I would have some water at the pool (or in T1) for when I exit, since I get a LOT hotter pool swimming versus OWS.
For the bike, assuming there are no aid stations, I would have a 20oz water, and 20oz sports drink mix in my cage.
For the run, I'd probably carry a bottle of sports mix.

Always carry a bottle on the bike

Anything over an hour on the run

And always take in water on the swim (unintentionally)

Always H2O with somewhat dilute Skratch Labs) on the bike and plain H2O on the run.  My mouth gets really dry so I have a small hand-held bottle I use on runs.  About 1/2 of each gulp gets spit out, the other half goes down the hatch.  I overheat pretty easily so I use the H2O at the race aid stations to dump on myself to help with cooling.  I also keep a bottle of Skratch at my transition area so I can take a few sips of that during transitions. 

x2, and I will add calories (either sports drink or gel, etc) if it's over 90 min.  For a sprint tri, I usually have a sports drink bottle on the bike and just water from aid station on the run.

Training:

Bike: one bottle per hour.

Run:  I've tried a carry bottle or hydration belt for runs > 1 hour, but found they were too much of a hassle.  I now plan long run routes around public water fountains, and carry a few buck so I can stop at a convenience store to get something if need be.

Swim:  I normally have a water bottle at the end of my lane for drinks between long sets.

Racing: 

One bottle on the bike through Oly distance.  On the run I rely on the course water stations.

Mark 

Swim - if doing a hard workout that will take over an hour, never in a race

Bike - always, unless very short race

Run - depends on conditions, can go without for 2hrs+ when cool & dry, but never carry in a race (use aid stations)

To forgo drinking in a tri, the race can't take me much over an hour and temps have to be favorable (ie, cool).  Would not drink in transition (better to drink on bike or run).

For your tri, I would probably put a bottle of water or sports drink (probably the latter) on the bike.  Maybe I would consider not carrying any and getting something from an aid station (if they have them) early on the run, if it seemed cool enough.  If there are no run aid stations, definitely carry the bottle on the bike and drink from it.  Then you won't need to carry anything when you run.

On the bike i will always have at least one bottle, if short distance then it will be water.  for training runs it depends on the distance and pace.  typically over 1 hr 30 mins i will wear my fuel belt or if im doing a harder longer tempo of an hour i will wear the fuel belt then as well.

for a race:  again on the bike same as above but for the run it depends on where the aid stations are and how far apart they are.  There is a marathon for example, up north of where i live that has an aid station every mile starting at the 1.5 mile mark.  For that event i don't wear the fuel belt and take sips of water or gatorade from the aid stations.  There i another marathon in my town (about 10 days away now!) that only has 13 aid stations and are not evenly spaced apart so i might be wearing my fuelt belt.  For ultras i always wear the fuel belt or camel back because those aid stations are a lot farther apart!

I always carry at least one bottle with me on the bike in training.  I've done rides of around an hour where I didn't drink if it wasn't hot, but it's nice to have it there since it doesn't make any difference (in training) if there was no bottle at all.

For shorter sprint like races, I'll usually have a small bottle that is half full with sports drink.  I'd say about half the time I don't even touch it, but sometimes it's nice to have just to wet your mouth.  Also, some say that the sugar taste in your mouth sends some type of signal to your brain that allows you to push just a tad bit harder.  I'm no scientist to know whether it's true or not, but having a small bottle between my aerobars shouldn't hurt me aerodynamically (and may actually help), so I don't see what the downside is.

For the run, I would just sip off water or sports drink at the aid station(s).