Fishing is very popular in the Dominican Republic.
If you are interested and located on the North coast, you can try it in Sosua (Puerto Plata province). There are many companies offering various boat trips, amongst them also fishing. (Most preferred way is Trolling. It is fishing style where you cast a line that moves slowly, for example from running boat).
You make a deal at least day in advance with them and you just show up in the morning at the parking lot and hop on the boat. They have everything prepared- fishing rods, baits and also beverages. So, it is also recommended for total begineers. Do NOT worry :-)
If you are lucky, you can catch beautiful fish. After you come back, you take your "trophies" home for consumation.
One of the most valuable types are tuna and Mahi Mahi (in Spanish it's called Dorado). We made tasty Ceviche salad from Mahi Mahi.
We also caught Baracuda, but captain did not recommend to consume it. It can be dangerous because baracudas living around coral reefs eat venomous fish like Blowfish.