Not many touristy things to do during the day, besides the monument, there's Centro Leon, a museum and cultural center. Walking around downtown (Calle Del Sol area) is pretty cool, Also, near the monument is the Teatro de Cibao, occasionally some great events there. Camp David is cool to check out for lunch or drinks, especially at sunset, the view is awesome. Plaza Internacional can be decent for people watching, Las Colinas is ok sometimes, Bella Terra is only decent at night, empty during the day. All three malls have great theaters with English language first run movies for about 1/3 the price of the US...
At night there are dozens of discos (with "non-working" girls), after hours clubs, tipico bars, car wash discos (with working girls), trendy cafes, decent to great restaurants...also a few areas with food trucks that get packed, especially on the weekends. There are two decent sized Casinos (Gran Almirante, Matum). There are big name (for the DR) concerts every weekend at clubs and bars around the city...just keep an eye out for posters.
Tip: in the DR "Night Club" usually means an adult club, like a strip club, although some of the biggest concerts are held at these places. What Americans call a night club is called a Disco or Discoteca".
Also, every Sunday in February, the Carnival parade takes over the monument area and Av. Las Carreras, family-oriented during the day (definitely worth seeing), but gets full of rowdy drunken teenagers once the sun goes down.
Do some searches here on DR1 and you'll find plenty of addition info on specific bars and restaurants.
