Hi back, Mom!
Wow, what a great birthday gift! You will have a wonderful time!
We hope to leave out of Boston on the 17th of July and return on the
30th. The cruise departs on the 21st out of civitavecchia and returns
on the 28th.
Here's the skinny, as I see it, anyway. You don't have really time for Amalfi on this trip unless you put it on the front end, and you'd have precious little for Rome if you put that one on the back end: just one full day. It needs a minimum of 4 nights/3.5 days to even scratch the surface without dashing madly from one attraction to the next.
Given the time it takes to get down to the coast and back to Rome to fly home, you definitely can't do it on the back end, and I wouldn't do it on the front either. It would be an exhausting trip on arrival day, and you'd only have 2 full days for the Coast, Capri, Pompeii, etc. before having to turn around and travel to Civitavecchia.
What I'd do? Scrap the coast and see if you can't add a day or two to the front end for Rome. As hot and as crowded as it will be - and with young children along - being able to sightsee at a more leisurely pace will help all of you be able to appreciate it more. Oftentimes travelers who've said they didn't care for it tried to cram everything into 2 days or so and ended up overheated and exhausted from being in the thick of the masses all the time!
Concentrating on the Eternal City will also eliminate the stress & expense of trying to move 9 people to/around the coast and back in a short period of time during its busiest season. Get into Rome and settle in; spend the rest of the day just getting your bearings; sightsee for the next 3; then head off the ship without the somewhat arduous journey it would otherwise be from the Amalfi.
Pickpockets: yes, they're a reality but not just on transport. You're going to want to protect your valuables ANYWHERE you are. More ways of doing that have been discussed on these forums than I can shake a stick at so you'll want to do some reading up to see which of those will work best for you. The short of it is that you never, ever want to carry your cash and cards in an exterior pocket or a standard purse. Ever. They should be stowed under your clothing or in a secured, cross-body bag (we use a Pacsafe). If using a bag, you carry ON you at all times: it's never laid at your feet, hung on the back of a chair or laid on a table. It's also a good idea to distribute the cards/cash amongst you so that they're not all in one place. I carry some cards/$$, and my husband does too.
LOL, we've had umpty lively discussions on 'best' methods of keeping the goodies away from light fingers, and no single way works for everyone so do some browsing around the forums for the various methods. The GOOD news is that the vast amount of tourists don't lose a thing if they've taken at least some sort of precaution, and even many travelers who haven't have escaped unscathed. It's best not to take the risk, though. It's no fun to spend precious sightseeing time/energy worrying about your stuff so just eliminate that particular distraction: problem solved. :O)