Hi, as this is your first time cruising I would maybe go for a Family friendly British ship, something like P&O's Ventura, although it has bad reviews this is in the main part from couples and people who have cruised before, we are travelling on her for the 1st time in 5 weeks but prior to this we didn't have children so opted for the ones that didn't encourage children with free places (Ventura offers children free in same cabin) the Ventura is geared up to entertain children and have 4 birth cabins aswell (two beds fold up out of the way on to the wall) also look at a cabin with a balcony bareing in mind you don't spend much time in them. You may also want to have a look at Ocean Village Cruises. Formal nights are really upto you and they tend to only happen twice on a two week cruise, attire tends to be jackets req by men and you girls will just dress in your favorite party gear (not jeans though), if you don't want to get dressed up just avoid the Dining rooms and opt for the buffet areas. If you fancy going further afield like the Caribbean, try Royal Caribbean (my favorite) but they will charge the same for Children or Ocean Village are worth a look aswell. AVOID the cheaper cruises like Thompsons etc, you will be put off for life!
I would ask yourself whats most important to you: Childrens entertainment? Food? Ports of call? what the ship looks like? etc.
Hope that helps a little. Regards.