There are lots of things that you can do together day & night. However, you will find like I did that everybody has different likes & dislikes so it may be hard to determine "family" things. As poster mentioned, the Capers are delivered to your door everyday so you can look at the next days activities. Bring different color highlighters for each of you so you can highlight the things you are interested in and see how many of you like the same things. There is family karaoke which happens a few times during a cruise which is in the bar next to the sports bar just forward from casino deck 2. I forgot the name of it. They also do adult karaoke in there in the evenings. The "shows" in the theater are set for everyone however, they have late night shows that are strickly adults cuz they can get pretty vulgar.
I'm not sure how old your kids are but just be aware that their 'family" shows can be pretty vulgar too. I was surprised at the language and content of one of the family shows where little kids were. My kids are older teens so it doesn't bother us any but for little ones, it is your call. Be prepared to walk out covering your childrens ears. I'm talking about the comedians and magicians shows. I don't know about the other stuff.
This is a very interesting site and toward the top is a link for capers which is the daily newsletter that tells everything happening on the ship which will give you some ideas of what they offer.
http://cruiseclues.com/shipstips/carnivalpride.htm