I would suggest:
Santo Domingo
1. National Aquarium on Avenida Espa?a. It's one of the finest aquariums in the Caribbean. It even has a plexiglass tunnel where visitors walk through while sharks and other sea creatures swim "above" and on the "side" of you. I'm not too sure if it might scare the really young ones, but for all other kids and those who are kids at heart, the aquarium is always a treat! There is an orphaned manatee that was rescued off the coast of Barahona a few years ago and now is being taken care of at the aquarium and has become quite famous!
2. Agua Splash Caribe also on Avenida Espa?a. It's a water theme park with plenty of slides and pools to spend a good afternoon. The is VERY popular with locals on weekends.
3. National Zoo on Avenida de Los Reyes Catolicos. It's one of the largest zoo's in Latin America, it's actually extremely humane. (ie. many of the animals are enclosed in open spaces and attempts to recreate their natural habitats are noticeable - many African animals). If you want to see the hutia or the soledonon, this is the place since these Hispaniola natives are probably extinct in the wild. There is a train that shuttles visitors around, but with such nice gardens and the relatively low number of visitors, why not enjoy this "African adventure" in the middle of the largest metropolis in the Caribbean with a leasurely stroll?!
4. Horse drawn carriages are found in Santiago and in Santo Domingo. What kid don't like horse drawn carriages?!
5. There is a new children's museum in the Zona Colonial. It's one of the newest museums in one of the oldest buildings in the hemisphere (from 1500s), is that odd or what! I really can't say what this place offers since I have not visited the place, but sinces its clear from its sign that its a children's museum, it merits attention here!
Puerto Plata Area:
1. Columbus Aquapark on Carretera Luperon before entering Sosua. Water theme park, the equivalent to Aqua Splash but in the north coast. Columbus is the better of the two in my opinion, but then again, it's only an opinion.
2. Ocean World in Cofresi. The best marine theme park in all of Latin America! You and the kids can swim with dolphins, explore artificial reefs, feed sea lions, and get close to sharks. You can even feed a tiger, behind an extremely thick window with a hole in it, there, don't want you to think that feeding the tiger involves loosing a body part or two, kidding! Overall, this is an impressive place, nothing like this exist in the Caribbean, except maybe for Dolphin Encounters in Jamaica, but this one is much better!
Punta Cana
1. Manati Park. The same as Ocean World, but less impressive and slightly different. This is only worth visiting if you will be in the Punta Cana area, since it's the only attraction (besides the beach and a go-karts place) in that mega tourist zone. Not to mention that the place is miles from anywhere, other than its own self contained self!
As far as I'm concerned, these are the ultimate kids stuff and places on these eastern two-thirds of Hispaniola. Of course, if the kids are much more mature mentally and might enjoy history, then this is the right place, being the first spot Europeans colonized. We have a historic thing or two or three or four, etc....
Enjoy!