I will be taking our 5 mos old in Dec (delivering tomorrow), but have taken our son now 3.5 yrs on several cruises (11 mos, 23 mos, 35 mos). From what Ive seen regarding little ones-I really watched them last time since I was preggo (there was a Brittish couple with a 2 week old last time):
I have seen the travel system strollers on board but it looked pretty awkward pushing it around, especially in crowds, narrow hallways, and forget about getting into an elevator during rush periods. Something smaller would be a lot easier to deal with and store in your cabin. HAve you thought about a carrier? I ordered 2 Hotslings and plan on using them a lot. Many times we folded up the stroller and carried it up stairs to avoid the elevators. However, you dont need to go all the way down to an umbrella. They do make smaller, light weight strollers that recline (a must for infants since they nap a lot). Check out Maclarens (really nice, but $$$) or something like the Graco Lite line. You could probably just wheel it up to your table during dinner without the car seat.
Ive never heard anything about not breastfeeding on board. As long as you do it in private you shouldnt have any problems. I will be nursing as well. I dont see why you couldnt even be sitting at the pool with a towel covering you up and nursing. I hardly notice people nursing in public most times and are amazed at how easy they make it look. I nursed my son and it took a while to be graceful about it.
Be sure to pack lots of diaper supplies, baby meds (Tylenol, Motrin, etc), first aid supplies, and any other items that you use a lot. The ship will most likely not have baby stuff on board, and if they do, it will be $$$$. Plus, you dont want to be scouring ports looking for items either.
One last note....you may have read many negative posts about bringing small children on board. Ignore them. We relish cruising since it makes us be a family. No phones, no work or housework, etc to deal with. My husband comes home a new person every time. Just be respectful around others. If your baby starts crying during a show or dinner, take him out and calm him down. Our waiters have always been wonderful and will either send the rest of my meal to the cabin, or I will just order room service (only had to do it once when my son got a nasty ear infection). Change his diapers in your cabin (ask the attendant to give you extra plastic bags and to clean out the trash can more often-theyre understanding. On the ships Ive been on, they didnt have changing stations in the bathrooms so our cabin was the best spot. Skip excursions and do stuff on your own. Theres nothing worse than to be stuck on a tour that you cant get off of when your child has had enough. We usually walk around port, shop, and then head to the beach for a few hours. I also tell my husband that if he wants to do something to go ahead. I dont mind staying behind. Have a great time and be sure to take lots of picutures for his scrapbook. Our son loves looking at all his pictures!