The traditional foods of Panama are inspired by its Spanish roots: lots of fried food, in general. The most important dish is SANCOCHO, which is basically a very tasty chicken soup, but there is more to that... especially if you have not yet discovered the subtle flavor of the "culantro" leaf. But of course, where you come from, chicken soup does not feature yam ("ñame"). And free-range chicken ("gallina de patio") is best!
The best traditional foods are usually found in the country's "interior". I'm a bit partial to the Azuero Peninsula where I lived and where I've come to appreciate the "fondas", rather rustic and friendly food places, usually found on the side of the main road, under roofs made of palm leaves. There you can find "pastelitos de maís nuevo" (made from fresh corn, ground on location) and "tamales" (soft corn paste, mixed with meat, wrapped and cooked in a banana-tree leaf), "arroz con pollo" (rice with chicken), "empanadas" (which my dictionary translates as turnovers), and all kinds of fried delicacies, such as "carimañolas", which are as good as they are cheap.
For a country nested between two oceans, seafood ("marisco") is far from being prevalent, if you except the ubiquitous "corvina" (which you could translate as some sort of Chilean bass) and the "pulpo" (octopus). Be sure to try the "ceviche" (raw seafood "cooked" in lemon, with plenty of onions), the perfect appetizer.