Try to relax, it's much easier than it might seem. You do tend to be on edge the first time or two checking into/boarding a cruise ship. It certainly can be intimidating. Not to worry, just think of it as checking into a hotel and a flight at the airport at the same time, that will really help. Put all your cruise docs and passports etc. in an easily accessible pouch.
I would also reccommend that before you leave home be sure that you purchase some durable luggage tags and fill them out with your cell phone number, (assuming you'll have your cell phone with you). Believe me, I've lost luggage on a few cruises and wish someone had told me this. The cruise line will give you color coded luggage tags but these sometimes get ripped and removed accidentally. Also another tip regarding luggage... when you depart they load bags for all the departing guest in huge bunches. It's hard to find your luggage sometimes. I've had people grab my bag a few times by mistake. I would make some kind of very visable attachment, like a colorful hair scrunchy or a pom pom yarn ball or anything that will clearly differentiate your bags visibly to you and to others.
For your children. Be sure to check them into the children's programs early so they can start making friends right away, otherwise they'll be too shy to join in mid way.
Checking out is easier, I think, with the exception of packing the night before. I start organizing a few nights before so I don't have to spend 1/2 that night freaking out about getting packed up.
Most of the muster drills are no more than 30 mins. You'll be in your cabin by then. I reccommend not being the first one's out when they make the announcement, wait a few minutes. It'll make it go quicker for you with the kids cause when you get down there you'll have to stand there for a while while waiting for everyone. Maybe not a good idea to pass this
tip out to too many people or the drills may end up taking longer LOL . Certainly don't be the last one's out or you'll get dirty looks from the other passengers.
Remember to relax though. Getting stressed about this stuff can make your first day tense. It's your vacation. It really isn't that difficult. Boarding/ticketing can seem like it's confusing, lots of lines and people every where. We usually board about an hour before the ship leaves ( only because we're always running late! we live in Orlando and drive to the cruise terminals). I would get on board as soon as they allow you, usually 1:00 pm or so on most ships. Carry a carry-on with you with anything you'll need for the first few hours on the ship because you won't see your luggage til much later. Go on deck for the sail away party to set your mood right, lots of fun and it will set the mood for the rest of the vacation! Have a great time!