Yes you can get around without a car, locals do it all the time, taxis in the city and towns, buses between cities.
Skip staying at Gamboa, take a day trip out there but stay the extra nights in PC.
If you know where you want to go and what you want to see,than taking a taxi is the economical way to go $2.00 US to anywhere in the city. If not contact a tour guide service a number of which are recommended elsewhere on this site,they will speak english and save you time and offer convenience,
Guys, the OP is only planning a few days in PC. True, no need for taxis in the city, but a taxi from David to Boquete will be kind of spendy. I'd rent a car in David, drive to Boquete and your other stops, then return it in David. (I assume you're flying back from David to PC; one-way car rentals down here tend to be very expensive.) Alternative: Take one of the frequent and cheap buses to Boquete, then rent the car after you come down the hill. But, personally, I'd prefer to have a car in Boquete to poke around freely.
Another reason to skip Gamboa: The taxi fare out there is steep.
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