Like you my first cruise was on the Triumph last year. It's a huge ship and it will probably take you the whole trip to find yourself around. Make sure to keep a map of the ship with you or check it before you leave your room. There is one pool on the very front part of the ship that's pretty secluded and I never found it. I know it's there cause it's in the picture but my friend and I couldn't find a way to it. You might be able to get there through the Rome Lounge. There are stairs on each side of the stage that go somewhere, try them. It's probably a good place to relax without all the kids.
I don't know how many people you'll have on board but as a note of warning make sure to get a lounge chair early if you plan on laying out or just relaxing by the pool. I went last year over Easter break and we could never find deck chairs on our days at sea because the families would save chairs for their kids that would never use them, and carnival couldn't enforce their no saving policy.
As for food, you have to try the Noodle Factory in the cafe (with the buffet line). It's on one side with the New York Deli on the other side. The food is very good there.
I wish you happy sailing and make sure that you have fun. Don't be too picky about the service and everything else. You're gonna find some gliches but if you just pass them by your trip will be the best one you take.