I recommend residency for sure if you are going to live in Costa Rica full time. The process can be long but worth it. You need your criminal record for where you have lived for the last 6 months and a birth certificate issued by the state you were born. They both need to be apostilled in the state where the documents originated from. Normally this is done by the secretary of state in person, mail, or on the internet. (Depends on the State)
An Apostille is a certificate issued by a designated authority in a country where the Hague Convention Abolishing the Requirement for Legalization of Foreign public documents.
You can make application in Costa Rica or at a consulate in the US. Not all Costa Rica consulates provide this service. I would wait until you arrive in Costa Rica to make the application with a qualified attorney. But beware of the time expiration of the documents. The documents are only valid for 6 months. So get them just before you arrive in Costa Rica or just before you start the process.
An Attorney will cost you about 1500-3000 depending on the type of residency and the attorney fees.
You will need to be fingerprinted in Costa Rica and passport sized pictures so that immigration can verify who you are and check your record with Interpol. Once you make the application you will be given a document that states you are in the process of gaining residency. This document will allow you to remain in Costa Rica until the application is approved. Once approve you will need to go to San Jose and pay a few hundred dollars in fees and they will issue your residency card.
My recommendation is to contact an attorney to help with the process. If you need a referral, let me know.