I am researching British warships which arrived in the Hawaiian Islands, particularly Hawaii Island a.k.a. The Big Island, in or about 1839. I am unable to find comprehensive lists of ships and crew. Where should I look? I want to learn the names of the crew members on each ship, from what British port they sailed, their mission, the dates they arrived, their impact on the islands and her people, the dates they left, and the names of those who did not return with the ships when they left.
If you want that level of detail then you'll need to work your way through the British Admiralty archives.
Ship's muster books will give you the crew names and should show if any were missing once the ship left the islands. The ship's log books (both Captain's and Master's) should give you the arrival and departure dates and may also give some idea of the mission. It's also possible that official messages or letters sent from the ships to the Admiralty will give you some idea of how the ship's were received on the islands but often the descriptions of what happened in port only cover official political interactions rather than social ones.
As a starting point, you might want to get a copy of Rif Winfield's "British Warships in the Age of Sail, 1817-1863", which gives short service histories for all of the vessels in the British Navy during your period of interest, to help narrow down which ships might have visited Hawaii.