By all means take the kids! Ive taken my son when he was 11, 23, 25 and soon to be 47 mos (and now our daughter who will be 5 mos)! I cant help with Carnival since we go RCI.
At the airport, you can wheel the stroller right up to the gate and gate check it (be sure to have id on it, and get a gate check stub). When you get off the plane, it will be right outside the plane's door. You will need to fold it/set it up, and be sure to remove any items from it. If you cant lock it shut, duct tape would probably work. If you have a bag, they may make you check it with luggage. If you do check it, get a bag since it will protect it from scratches, etc. However, you will want the stroller walking through the airport, especially if connections are tight. The stroller acts as a nice wheeled luggage cart if you end up carrying your kids. Same goes for the transfer-tell the driver you will need the stroller as soon as you get off the bus. Hopefully he will put it in last so it can be retrieved easily. Again, you will want the stroller for boarding and getting around.
Otherwise, if you leave it with luggage you may not see it until after dinner time. I was lucky that my son was walking at 9 mos so we didnt have to worry much about a stroller. Keep in mind too-any security areas you pass through (airport and ship) you will need to fold the stroller up and put it through the xray machine so be prepared before you get up there so you dont hold others up. If its too big to fit, they may make you check it through, or if youre lucky, you can have it hand scanned (if they still do that). I didnt use a big stroller-just an umbrella one with no car seat so I cant help much there.
Any other tips: be respectful of others. When your kids get fussy, remove them. Dont run over other people's feet (sometimes it's hard though when you get idiots who see you coming and jump in front of you. Then they give you a look like its your fault. The worst is people who walk side by side taking up the entire walkway). We were able to go to shows as our son often fell asleep. Just be sure to sit in the aisle where you can make a fast getaway if you need to. Bring a jacket or blanket for the cool nights/rooms. Be sure to bring a favorite toy, blankie, animal, etc to make the transistion easier. Arrange to have a crib and highchair before you sail just to be sure they have it on board. Usually the first night at dinner you have to ask for a highchair, but the rest of the time it will be waiting for you. In the other dining areas (cafe) you will need to get one every time (ask a waiter for help). Stick your stroller off to the side and out of the way when not in use, or leave it in the cabin to get later. Bring plenty of diapering supplies, bottles/sippy cups, dish soap and brush, favorite snacks/food, sunscreen, hats, cover ups, baby meds (Tylenol, Motrin), baby size first aid supplies, extra pacifiers if you use them, etc. They dont carry infant stuff in the ship stores and you will not want to run around port looking for items and paying high prices. Skip excursions, though your kids might still be young enough to sleep through the long tours. Some have a minimum age req anyways. The problem comes when they get fussy and you cant get back to the ship easily. Avoid the heat and sun during the height of the day, bring along water to keep them hydrated, and take a break before dinner so they can nap in the cabin (this is when a balcony cabin comes in hand so you can sit out there while the kids nap-course sometimes we nap too!). You will be thankful if they nap before dinner! Take advantage of the photo ops for nice family portraits too.
Have a great time, and if you want any more tips, read the gripes about bringing small kids on board. Dont let it scare you, most people are very accepting of small kids, but you will run into a few that will give you dirty looks, etc. Just smile back and go on your way. As long as youre not being disrespectful, they have no reason to complain.