We recommend to use private drivers or Uber rather to rent a car.
Driving in Panama is not easy as the trafic is not well marked and Ubers are very cheap.

PANAMA CITY 
Poin Panama www.poinpanama.com (zipline in the city)
Miraflores Locks (Panama Canal)

Amador Causeway (museum, kids park, bike rental, kosher icecream ...) 
Casco Viejo (old city)
Biosphere (museum)
Panama Viejo (antic Panama)
Ancon Hill (hike to the highest point in the City)
Rent bikes along the water
MultiPlaza (high-end mall)
Parque Natural Metropolitano
indoor kids parcs (crazy park, fun spot, game world...)
 
OUTSIDE THE CITY 
Anton Valley (hiking, waterfalls, horseback riding)
San Blas islands (paradise archipelle with hundreds of paradise islands)
Monkey Island boat tour
Rainforest Gondola tour
Gamboa bike park (bike in the jungle)
Contadora (Pearl Islands) - whale watching, snorkeling/scuba on Chipera, Mogu Mogu, https://pearlislandsdaytours.com/
Rainbow bass fishing
Ziplining
Rainforest
Beach- Santa Clara and Punta Palmar both 1.5+ hours from Panama City. Buenaventura resort
Embera village (Indigenous tribes)
Coffee plantations in the mountains
Whale watching
Chargres River Tour, hike to waterfall
Hiking Portabello road
Boquete (chabad Shliach here)
Bocas del Toro (chabad Shliach here)