I strongly recommend Princess. My personal favorites are Celebrity and Princess, but Celebrity tends to attract an older demographic and can, at times, feel a bit stuffy. Princess is a happy medium between luxury and casual family fare. On their newer ships there are ample-sized children's play rooms, children's activities leaders, and most of all, other children to play with! Plus, you'll have the company of other parents around your age. Princess treats their guests pretty royally (no pun intended...), while still maintaining a mostly casual atmosphere around the ship.
If you want a more family-geared vacation, aim for Disney Cruise Line, who magically has managed to separate the adult and child cruising experience. While there may not be a casino, and you will encounter mickey mouse EVERYWHERE, if you like that stuff it's still a great experience and they know how to pamper adults. Their specialty restaurant, Palo, is particularly worth trying (it's adults only and has a surcharge).
Good luck, and have fun! |