Go to a college or university library. Although some public libraries may carry peer-refereed journals, your best choice is an academic setting. Peer refereed journals tend have rather esoteric articles, so they don't generally appeal to a wider reading audience as the scholarly language assumes that the reader has a working knowledge of the discipline.
Check with a librarian. People who work in libraries are often knowledgeable about the different peer-refereed journals available in the different disciplines. They may have a list of journals available in the library that you can refer to.
Consult Ulrich's International Periodicals Directory, which should be available at the reference desk. Go to the "Refereed Serials" section of the resource and review the journals listed there. Cross reference it with the peer-refereed journals that are available at the library.
Ask your professor or adviser. If you are an undergraduate ask the individuals whose classes you are taking the names of the peer-refereed publications they recommend. If you are working on a master's degree of a PH.D your adviser would be aware of the peer-refereed journals.
Conduct online searches. CalPoly recommends that you consult the Internet according to your area of expertise. If, for instance, you want peer-refereed journals in education or the social sciences, check ECONbase or Project MUSE. For science, check with Science Direct or ACS Publications for full-text articles.