FamilySearch and the LDS church have lots of Fijian records on microfilm. On the downside, they are not indexed or digitized yet (i.e. not searchable online). On the plus side, you can check out most of it to be sent to your nearest family history library to look through.
I have a few names and dates to start off my search, so I have ordered 4 rolls of film (2 births, 1 marriage, and a book on Indian migrants in Fiji). I hope I will be able to find the people I'm looking for and maybe enough info to need to order more film.
I hope one day they will get around to digitizing these so I won't have to order stuff and go to a library with a microfilm reader to look through it. It can be difficult to get a babysitter midday to drive to another city to sit and look through film. Oh well, I'm sure the work will be worth it in the end.