To obtain a legal name change in Rhode Island, an applicant must submit a notarized name change petition to the court. Along with the petition, the applicant must submit a certified copy of their birth certificate and a copy of their Bureau of Criminal Identification records (follow the instructions for state background checks here). You can request waiver of the publication requirement or publish notice of the hearing in a local newspaper once a week for two weeks (use this form). The advertising requirements may differ by county. (R.I. Gen. Laws Ann. §§ 33-22-11 & 33-22-28).
For instructions on legal name changes for minors under 18 in Rhode Island, see NCTE's Name Changes for Minors in Rhode Island resource.