Hi Carol,
The police department will notarize it there. The apostille will come from a “Competent Authority” for us this was the Secretary of State in Texas. See this website – https://travel.state.gov/content/travel/en/legal-considerations/judicial/authentication-of-documents/apostille-requirements.html.
As for all the documents you need, see this website – http://www.arcr.net/residency/requireddocuments.html. Requirements change seemingly with whim a good attorney will help navigate the process, although you can do it yourself for much less if your Spanish skills are good or you have an interpreter.
Finally, note that the date of the notarization or apostille is important. The document can not be more than 6 months from being verified. For example, if your certified birth certificate has been sitting in a drawer since you were young it would be invalid for residency purposes. You would need to go to county records and purchase a new one… then hurry to Costa Rica to file it 🙂
Best of luck,
Greg Seymour