My girlfriend and I did our first cruise a couple weeks ago (well, she did one with her friends a few years back before we met, but this was my first cruise). We chose Royal Caribbean "Liberty of the Seas" on a 5-day Western Caribbean itinerary out of FLL, calling at Labadee (Haiti) and Falmouth (Jamaica).
Royal Caribbean was probably a good cruise line to start out on - a step above Carnival, but not too expensive just in case we didn't like it. We had a balcony for about US$1500, all in. Obviously specialty meals, drinks, excursions were more. But for a 5-night balcony, I'd say we got a good deal.
Royal had good and bad aspects. Good - ship was lovely, crew were mostly all fantastic, room was larger than we expected, drinks slightly cheaper than we expected (note: not cheap, but cheaper). Poor - food was awful in most places on the ship, Falmouth (a private Royal port) was a pure money grab.
I still say Royal was a good one to do first.