I wouldn't recommend the zoo to someone coming from a first world country as they'll probably be quite disappointed, but it's decent by Dominican standards. The botanical gardens are nice if you're into plants, flowers, etc...
Cock fighting would be different I'm sure, but unless you really are into that "sport" I wouldn't dedicate more than a couple of hours to it. There are cock fighting rings in almost every town and coliseums in certain larger towns as mentioned above. Be aware that just like playing poker if you don't know anything about how things work the serious players who are betting real big time money wouldn't want a "green" person seating next to them asking questions and disturbing their concentration. It doesn't mean you shouldn't go, but I would go with a Dominican who could ascertain these things or keep myself just to watching the fights...
It all depends on what kinds of things you enjoy of course and if you know Spanish. Taking a dinner at Plaza de Espana is always a good bet for a tourist. if you're a fan of music go to a Bachata / Merengue, etc...concert or party (There are always some going on) and Visit Avenida Venezuela on a weekend to see how real (lower to middle class) Dominicans party it up. Visit "El Conuco" restaurant (a little touristy, but a little different experience). Take day or two day's trips outside the Capital to places like Jarabacoa and see the waterfalls, Bani and visit the dunes, Cotui and visit Hatillo Lake, etc...go to some river's "balneario" (i.e. Masipedro in Bonao, La Confluencia in Jarabacoa, la Toma in San cristobal) and spend the day eating, drinking, bathing in the rivers.
Beaches up the wazoo of course all over the country...but not in Santo Domingo...go to Las Terrenas and spend the day on one of its wonderful beaches. Eat lunch and/or dinner at one of the many restaurants by the seaside. You can also visit one of the other many beaches around the area including "Playa Rincon" - At one point rated one of the top beaches in the world. If you feel a little bit more adventurous hike up to "El Limon" waterfalls...
If you're into history The Colonial Zone is choke full of historical placest to begin with (There are many small museums around that area), but you can also visit "El Faro a Colon" where supposedly Christopher Columbus is buried, The museum of the Dominican man, or the museum of natural history, etc...
Go east to Boca Chica, Juan Dolio, Guayacanes or Bayahibe...take your pick depending on whether you want a large beach or smaller more intimate one.
Visit one of our national parks (i.e. Los Haitises).
If you want different go to a "Carwash"...not the kind you're used to...