There are two places in the Panama area of San Francisco that I can suggest if you are interested in rivers, lakes or waterfalls. You can either go to El Valle, which is two hours away from Panama City, or you can go to Gamboa, which is less than an hour away depending on traffic.
In El Valle, you can visit the iconic Chorro de las Mozas (Young Women's Falls) and the Chorro El Macho (Manly Waterfalls), the entrance to which is pictured here.
If you don’t want to travel the two hours to get to El Valle, a good alternative is Gamboa. It is basically tucked up into the jungle and is very close to the Panama Canal. Nature-lovers visit Gamboa for its tropical forest and diverse wildlife. You can take a boat tour from Gamboa towards Gatun Lake if you want to go fishing or see some birds and crocodiles.