Things to See:

Playa Cholon (Rent a boat and go to Cholon Island)
Castillo San Felipe de Barajas
Mercado Bazurto
Walk around the Walled City
Take a trip to Playa de Cholon or the Rosario Islands or both (Rent a boat with friends) [I prefer Cholon, you pull up with other boats and party].
Convento de la Popa (amazing view of Cartagena, the closest equivalent is going to Christ the Redeemer in Rio [better view])
Walk around in Getsemani and see the Graffiti
Go chill on the beach in Bocagrande

Restaurants:

Carmen
Donjuan
La Vitrola
La Cevicheria (amazing Peruvian Ceviche, try the lobster ceviche)
Alma
Mistura

Bars:

Cafe Havana (go salsa dancing here)
Alquímico
Mister Babilla
KGB (Touristy, but cool for a quick drink, Soviet Themed [as you might guess])

Rooftop Bars:

Cafe del Mar (Have drinks at sunset here, touristy but still cool)
Mirado Gastro Bar
Go to the Rooftop Bar at the Hotel Movich (best view in the city)
51 Sky Bar (in Bocagrande, very Miami-ish)

Clubs:

Bazurto Social Club
Media Luna (it’s a hostel but also turns in to a club)