GW – A great alternative is to ship a “crate” – you can go by air cargo or by taking up just some space in a container. You might try pricing out that cost versus shipping some stuff by mail and taking some in your suitcases – maxing your baggage allowance. It might turn out cheaper to pay the extra bag fee.
Christine – Any moving company you contact will be shipping your stuff down by boat. They pick up your stuff in North American and take it to a port in the U.S. like Miami. Then a Costa Rican moving company takes over when it gets here. But I’ll recommend that you first find a CR moving company and ask which U.S. company they want to work with because the CR company will be familiar with the customs process down here.
Dog and cat transport is easy. I wrote an article on this, check it out here: https://internationalliving.com/2012/06/how-to-travel-with-pets-to-costa-rica/