Of course, it´s an easy process! Many foreigners relocate in Costa Rica and bring their pets along with them! Some of them even go back and forth many times during the year between their former country and Costa Rica! All you need to do is get a current health certificate from your Vet, that is issued less than 10 days prior to your flight, and bring it with you. There is no quarantine of any kind. Most of the major airlines allow pets to fly.
The Costa Ricans overall, are huge animal lovers, and most of your maids and caretakers will treat them like family. If you have to go out of town and leave your furry friend, there is a wonderful couple who lives here in Papagayo, George and Kathy. They will take extra special care of your babies.Their website is http://www.guanacaste-animalcare.com/home.html.
Unique animals, like exotics or horses, need special permits and approvals so you should check with a Costa Rican attorney about the qualifications for bringing them.