Find Your Cruise
You obvious need to research where you want to go and look at the various itineraries. You don't have to do formal night on any cruise lines but on Princess will only be allowed to dine in the buffet for those nights. You can opt for anytime dining with princess so you don't have a set time or table and you can choose to eat alone or share a table with others, pretty much whatever you want to do. As for the children thing don't sail during school holidays.
You could look at Norwegian Cruise Lines (NCL) or Thomsons they will have the Island cruise ships under their wing shortly. If you pick a ship with a number of places to eat even on a formal night you do your thing and go to one of these for your food. On a 14 day cruise probably on 2/3 formal nights / on a 7 day cruise only 2 formal nights so not a major problem. It was always the intention of crusing to have formal nights being part of their tradition and something different you wouldn't likely do if you spent 2 weeks in Majorca.
Ocean Village, Island Cruises are more informal, although popular with families. However most cruise lines now allow you to dine wherever and whenever you choose without the hassle of 'dressing up'.
Princess (as Pilgrim states) have 'Anytime Dining' and although they have formal nights, you don't have to dress up for them. We always missed one formal night on board and chose to eat in the 24 hour buffet instead. (Besides, as long as you are not wearing denim, shorts or vest tops, you should be fine in the dining rooms).
P & O have an adults only ship. Stay clear of larger resort ships that cater for 3000+ passengers (Independence, Ventura, Princess ships; Crown, Ruby, Caribbean, Emerald) as these have far more in child facilities and cater more for families and those with never ending energy levels! Choose a ship with fewer than 2000 passengers as they do not have much to offer for the younger guests!
We went with Sea Princess during Easter 2008 (another booked for Easter 2010) and being a Primary School Teacher, I want a break from children. Although children were on board, there were fewer than on the Crown during Easter 2007, well behaved and very polite.
If you are prepared to spend a little more and have a really good standard with v. few children try Regent Seven Seas , seabourne or maybe one of the "sailing ships" of Windstar cruises .
You don't give your ages and it makes a big difference to the ships you might like. Smaller ships e.g. Fred Olsen's, Ocean Majesty, Calypso (with Thompsons last year) tend to have an older clientelle so if you are under 50 you'd be better with a bigger ship with a wider age group. With cruising you pay your money and take your choice. If a bargain deal looks too good to be true, it probably is!