The three biggest annoying pests in Panama are in no particular order:
Bad Scooter Riders
People Who Loudly Discuss Their Gastrointestinal Diseases In Restaurants
But enough about human pests. Let's talk about the kind of insect pests you'll encounter in Panama.
1. Fire Ants
Most likely encountered in: cities and homes.
While Panama is host to many ant species, the Panamanian Fire Ant is the one best adapted to plaguing heavily inhabited areas, stealing food, stinging folk and generally proving to be poor house guests. Their sting, while not dangerous (unless you are allergic) is still irritating and the welts they leave behind are itchy enough to make their visit an unpleasant one.
For the welts, I often use zinc oxide cream to alleviate the annoying itching.
To get rid of them, I suggest scrubbing their trail with detergent. Without a scent trail to follow they will not repeat the same path again. But to really end their visit, you have to find their nest and hand an eviction notice to their queen (if it's on an area away from my lawn, I usually just pour a pot of boiling water on the mound. In my lawn, I usually use boric acid crystals arranged around the mound).
Most likely encountered in: wetlands, jungle
The mosquito has been a bane of Panamanian living for centuries. In fact, much of the difficulty in building the Panama Canal was because of the Dengue and Malaria these aerial annoyances like to carry. Fortunately, the Panamanian people aren't ones to falter under insect invasion, and as such have been regularly spraying, drying areas, and otherwise driving the bugs off. As of this date, it is hard to find a mosquito in a city.
Zinc oxide for the bite. Coconut oil as a repellant (odd but true, pure coconut oil seems to annoy the bugs).
3. Sand Flies (aka Chitra)
Most likely encountered in: beaches, islands
These little, hard to see pests bite hard and bite often. If you have beachside property you will learn to loathe the life out of these tiny terrors. Fortunately, they fear fire and smoke more than most flying pests (so if you ever plan to have a beach BBQ, you'll find that the chitras will give you a wide berth.
Benadryl for the bite.
To repel them, I usually mix up baby oil, citronella oil, and alcohol hand sanitizer.