Don't assume you are in "a paradise". You are not. The DR is still a third world country with all the issues and challenges that come along with that. Everywhere is not safe. And if it is not safe, then it is not pleasant.
Take some time and read the Santo Domingo forum posts here on DR1. You will get an idea for the flavor of the city, the good and the bad. Keep your head up and keep your stick on the ice.
Not knowing your tastes or what you would be looking for, I can try to make a few more suggestions for safe and pleasant excusions.
The museums at the Plaza De Cultura. There are several of them ranging from natural history to art.
http://www.tripadvisor.com/Attracti...anto_Domingo_Province_Dominican_Republic.html
The Museum of Resistance in the Zona Colonial is very interesting.
http://www.tripadvisor.com/Attracti...mingo_Santo_Domingo_Province_Dominican_R.html
The Folklore shows on the Plaza Espana in the Zona Colonial on Saturday night. It is a great place to people watch and have a cold drink in one of the outdoor cafes or restaurants.
http://www.fodors.com/world/caribbe...s-to-do/sights/reviews/plaza-de-espana-451689
The music at the San Francisco Ruins in the Zona Colonial on Sunday night.
A very good guide to the Zona Colonia.
http://www.colonialzone-dr.com/
At the other end of the city there is good jazz at the Dominican Fiesta Hotel on Friday nights.
As mentioned the Botanical Gardens are beautiful and pleasant, though they are very dry right now.
http://www.jbn.gob.do/
And there is bowling!! What Fun!!
http://sebelenbowling.com/
Outside the city you might consider:
A boat ride to Isla Catalina from La Romana.
Another cave park outside San Pedro. Very nice with guides, an easy walk and Elevators!
https://es.wikipedia.org/wiki/Parque_nacional_Cueva_de_las_Maravillas
If you have more time you might consider a trip to see the terrace farming and stunning scenery in Constanza and Jarabacoa. The new road that connects them is absolutely beautiful. It is a 2.5-3 hour ride from the capital.
Most of all, enjoy the time with your dad. Enjoy and give us a trip report when you return