I am going to weigh in to second the recommendation of Celebrity as one of the best mid-range budget lines. They are really, really great! I have cruised many of the major lines long ago and both Celebrity and Princess more recently. I would not hesitate to say Celebrity is the best line I have been on. I haven't tried the higher end lines just yet, but maybe soon. Shop around to find the best deals. It helps if you are flexible with your timing or destination to find a great cruise at the right price for you. If you have never cruised any line besides Carnival, you are in for a real treat. Carnival is the greyhound bus lines of the seas.
If you are looking to avoid tons of kids running around (how I wish I could!), consider taking a longer cruise if your budget and schedule allows it. The typically family only takes the kids along on a 7 day trip. Look for 10-12 days and you will have a great reduction in the number of kids on board.
Our most recent cruise was on Princess in January of last year. We went to Ocho Rios, Panama (including partial transit of the canal), Costa Rica, Cozumel, and Grand Cayman. Each port was great with the exception of Ocho Rios.
Research the ship that you are thinking of taking, and each of the destinations that you might be stopping at. Look into shore excursions on your own before the cruise. This alone can save you hundreds of dollars and ensure that you have a really great time. Avoid using shipboard excursions if at all possible. They are a rip-off and usually too short as well. A great resource for research is
CruiseMates.
Keep us posted on your selection. Good luck!