We have been on both lines, I love RCI, The Enchantment of the Seas is my favorite ship so far, and we were on that back in 2001. We were also on the Navigator, also beautiful. All the glass, and the adult pool area are great.
We have been on 2 Carnival ships, we were just on the Glory in May. I can't say enough good things about it. The food was a lot better on Carnival I thought. It was not the drunk party atmosphere everyone thinks. It was a beautiful ship filled with all kinds of people having fun.
Both lines are great, they both offer different things. I like the cabins on Carnival better, more space. I like the food and shows better on Carnival, I like the deck space/layout better on RCI. Prices are better on Carnival.
I have never been on the Triumph but I have heard great things about it, and the ports are wonderful! Go and have fun, you can't go wrong with Carnival. Give RCI a try next time!