From Panama City sits on the Bay of Panama, however the city itself doesn't have a really good beach, there are few small beaches but not frequently used. So we'll refer to the beach areas of:
PACIFIC BEACHES
Playa Bonita - Just across the Bridge of the Americas from Panama City
Playa Dorado - about 20 minutes from Panama City
Punta Chame - 1 hour from Panama City
Gorgona - 50 minutes from Panama City
Coronado - 55 minutes from Panama City
Punta Barco - 1 hour from Panama City
San Carlos - 1 hour from Panama City
Corona - 70 minutes from Panama City
Santa Clara - 80 minutes from Panama City
Playa Blanca - 90 minutes from Panama City
ATLANTIC BEACHES
Maria Chiquita - 40 minutes from Panama City
Portobelo - 1 hour from Panama City
PACIFIC BEACHES
Playa Bonita - Just across the Bridge of the Americas from Panama City
Playa Dorado - about 20 minutes from Panama City
Punta Chame - 1 hour from Panama City
Gorgona - 50 minutes from Panama City
Coronado - 55 minutes from Panama City
Punta Barco - 1 hour from Panama City
San Carlos - 1 hour from Panama City
Corona - 70 minutes from Panama City
Santa Clara - 80 minutes from Panama City
Playa Blanca - 90 minutes from Panama City
ATLANTIC BEACHES
Maria Chiquita - 40 minutes from Panama City
Portobelo - 1 hour from Panama City