Hello SciFiGeeks, and welcome to CruiseMates!
You have picked a good cruise for a first time, since it is long enough to give you an idea if you like cruising, and Carnival is a great choice for a family.
As to your question about ID: you do not have to carry passports with you off the ship in those ports. However photo IDs will be needed in addition to your ship's cruise cards in order to reboard. This is for the adults, meaning you, your wife, and the 18 year old. The other two kids will not need a photo ID, but if the 16 year-old has a driver permit or license I'd recommend he/she bring it ashore. Any sort of government issued photo ID will suffice. And by the way, kudos about getting the passports. You'll be glad you did it and not make the rookie mistake so many cruisers do, by avoiding the passport issue.