there is a fair amount of interesting climbing outside of the town of Boquete, several hours west of Panama City. the climbing is cool basalt formations with routes of all grades, though mostly lower end. the area is relatively temperate for the tropics, though as can be expected it is always quite humid. there are guide services in Boquete which you probably would need to hire for a belay as there are not really any "climbers" in the area, at least none that i ever saw. honestly, you'd be better off going on coffee and birding tours in Boquete or going rafting depending upon the season your there.
personally, i'd go surfing in Boca del Torro if you wanted to be cheap or in the Golfo de Chiriqui if you had some $$$ to throw down. Do not surf anywhere near Panama City, the water is NASTY! Veraguas has some great waves, but can become fairly territorial from what i hear. be sure to bring some tip top gear if you plan on going out on the town in the City, people dress to impress and like to party late. either way drink lots of Balboa beer, do yourself a favor and don't miss out on some Ron Abuelo 12 ano, and don't be afraid of the Secco Herrerano, it's cheap, looks like swill, the locals love it, and you will too.