Sounds like you need to do some research. Your library will have books on cruising that will describe the basics and the many, many options from which you will have to choose before you visit with a travel agent to book your cruise.
NCL bills itself as the "Freestyle Cruise". It's the lowest-quality line in terms of food and service, but it probably won't the the cheapest. It is very informal, which some people love and others hate. Having said that, all the cruise lines now have "anytime dining" which allows passengers to just walk in or make a reservation for a time that's convenient for them, rather than doing the traditional main/late seatings [although those are also available]. The other lines still have formal nights, but if you don't want to "do" formal, you can always dine in the Lido or other alternate venues.
Which ship would be "best" depends on a lot of factors - it's like asking which car is "best". What you need/want may not suit me and vice versa. That's why you need to do some research. Ships even vary within a particular cruise line.
As to your price limits, that may be hard to swing, especially if you're planning to go during school break [e.g. President's Week]. Air fare will also be higher at that time, so you need to consider that, too. In general, a cruise will cost about $100 per person per day on the mass market lines. In general, Carnival is a good first cruise, although not everyone likes the "party" atmosphere on their ships and the outlandish/garish decor.
Most of the Caribbean cruises will include a stop in either Nassau or Freeport [or a cruise line's private island in the Bahamas]. The Bahamas stops aren't that great, except for the private islands, which are terrific! You'll need to decide between the Western and Eastern itineraries and you'll need to consider how you'll do excursions in each port. Some are safer/easier to do on your own than others [e.g. Cozumel and Cayman are OK, Jamaica is definitely NOT]. Your library will also have books describing the various Caribbean ports. The website CruiseCritic.com has tons of current, reliable info on every aspect of cruising, although if you're totally unfamiliar with cruising, it might be a bit overwhelming at first. Click on "Cruise Styles" for a description of the demographics of the different lines, or "Reviews" for the different ships. Click on "Boards" for discussions on just about everything about cruising.
Good luck. Hope you find just the right cruise for you and your family. It's a really fun way to vacation!