After you and your same-sex partner have found representation -- preferably through an attorney who specializes in not only family law, but who is also familiar with the dissolution of same-sex unions -- you will need to serve your partner with divorce papers.
This is not something you will need to do personally. Your attorney should be able to arrange the serving. Be sure, however, that the documents are correct before having your partner served. Make an appointment with your attorney to go over all the papers beforehand.