We travelled with our four month old in December and we are heading off with her again in April (she'll be 7.5 months) and we have no issues. We travel with our own portable playpen so that she has some consistency in hotels, etc as we go but I know that the cruiselines do provide them if you need them. The fact that the baby can't go in the pools is always a bit of a disappointment, especially if you have a baby that loves the water but as others have said just bring a small inflatable pool.
We do take our kids to shows, including the baby but we are always mindful of those around us, when baby starts to fuss, it's time to go for a walk on the promenade or somewhere that is loud enough that a baby crying doesn't really make a difference. There will always be some people on cruises that think that they should be for adults only but they are few and far between, for the most part people warmly welcome little kids, especially those kids with considerate parents.
If I had to offer a few hints I would suggest that you make sure that you bring any medications baby may need, ie infant pain relief/fever reducer, teething relief, bum cream, etc. If you do manage to find these items on board or at the ports you pay a pretty steep price for them, better to travel prepared for anything and with the brands that you know and trust. And for your comfort and the comfort of those around you please evaluate the type of stroller you are bringing. With our little stroller we were fine, friends of ours who took their cadillac got a lot of dirty looks from others trying to share the elevators and hallways with the huge stroller. If you can handle it, we found the baby backpack to be the best way to get baby around the ship and off the ship without irritating any fellow travellers. Have a blast, these moments of travel will bring decades of wonderful memories for your family.
"Faith is to believe what we do not see; the reward of faith is to see what we believe."
-- Saint Augustine
Carnival Paradise 2000, Carnival Paradise 2003,
Carrnival Victory 2004, Carnival Valor 2005, Carnival Fantasy 2006, Carnival Fascination 2006, Carnival Destiny 2007
Carnival Sensation 2007. Carnival Fascination 2007 Carnival Triumph 2008, Carnival Fascination 2008, MSC Orchestra 2008, Carnival Fascination 2009, Carnival Glory 2010
Waiting for the Carnival Imagination June 28,2010