Just got back from cave tubing at Jaguar Paw where we stayed 2 nights. Staying over will allow you more flexibility in the cave trips. A day trip will limit your time and it is such a cool place we felt we could have easily stayed a few more days.
You will need shoes that can take you through jungle paths for about a one hour hike, then get completely wet while tubing. We got some shoes by Columbia called "river trainers" that were great. They are a kind of a light weight hiking boot that is made to wear wet. Got them at Boat US. Otherwise you could wear aqua shoes (the guides do) but you may find hiking in them difficult.
An old pair of sneakers might work, but the inside of the caves are very slippery in spots, and YES it is dark. Headlamps are provided so you will not be without light, but when you shut them off you will be in total darkness.
I can't think of a more fun thing for you and your kids to do, and it will give you a chance to see some of the mainland which should not be missed if this is your first trip.