Here are the primary sources for legal research in the State of Hawaii: statutes on the State Legislature's website, cases on the Judiciary's website, and administrative rules through the executive branch website.
Hawaii State Legislature’s website (link to statutes in in the left-hand column on their site): capitol.hawaii.gov
The Judiciary’s website: www.courts.state.hi.us
The State Executive branch administrative rules: http://ltgov.hawaii.gov/the-office/administrative-rules
Link to the county ordinances on the City & County of Honolulu’s website: http://www.honolulu.gov/ocs/roh/rohonline.html
While you can use the Judiciary's website to look for Hawaii cases, it is easier and faster to use a legal research database. The two primary paid legal research databases are Westlaw and Lexis-Nexis. There are several key videos on this page to help guide your research of you have access to Westlaw. If you do not, you can research for free on the Google Scholar site. Below is a video from the Library of Congress showing you how to search for states cases in Hawaii or states, or the federal courts.