I live on Oahu and am going to my property on the Big Island. I want to bring either my hang gun or rifle with me. What are the requirements behind that? Anything you can share is appreciated.

You have to check 2 things.

1, TSA rules via their website.
2. The airline rules of the airline you will be using.

There is nothing in state law you have to do if the guns are already legally owned.

Will check that out. Do I need to carry any paper work with me?
Mahalo.

Will check that out. Do I need to carry any paper work with me?
Mahalo.

There's only the declaration tag you sign at the airport.  The airline will provide that at the counter.

So, don't use a Sky Cap.  You have to have the gun at the counter to declare it.  Sign the declaration and place it inside the gun case and lock it.

TSA wants you to use TSA-accessible locks that allow them to open them without your key.  It's safer to use non-TSA-approved locks.  If they feel the need to open the case, they can page you.  I normally wait at the counter a good 5+ minutes to give the baggage time to clear TSA.  There are some airports on the mainland that routinely destroy non-TSA locks without first attempting to contact the traveler.   But, you are unlikely to experience that on an interisland flight.

I have flown the following airlines with Gun(s) and Ammo

Delta
Alaska
Hawaiian
Southwest

Pretty much what Flapp said is the same.  HOWEVER only one DELTA i have had to pick up my gun case/luggage from their baggage office.  Also now 2X they have TIE STRAPPED my luggage before handing to me.

BTW all airlines EXCEPT Alaska (Must be Alaska controlled flights and NOT their "contracted carrier) restricts Ammo to 11LBS.

I Flew interisland recetnly with ZERO issues on Hawaiian.

Aloha!

TSA Firearms rules which were updated recently:  https://www.tsa.gov/travel/transporting-firearms-and-ammunition

How I transport firearms:

1.  If flying to the mainland, check the laws of that state and any state you pass through.

2.  Lookup the airlines specific firearms policies on their website.

3.  Firearms, ammunition, and parts have to be in check in luggage.

4.  Verify guns are unloaded and in a locked case.  Put locks in all case holes and ensure you cannot pry the case open with your hands to remove the gun.  I recommend using regular locks but you can use TSA locks.  The case can be stored in luggage or on it's own.  The luggage the case is in doesn't have to be locked.

5.  Ammunition placed in manufacturers boxes or magazines.  You may need to tape the cardboard boxes closed if the boxes are worn.  TSA might reject loose rounds in manufacturers boxes or bags.  Most airlines allow for up to 11lbs of ammo per passenger.  Alaska Airlines allows up to 50lbs per passenger.  Ammunition can be stored in the firearms case.

6.  I recommend placing a tracker in the case, like an Apple Airtag.

7.  Arrive at the airport 20 minutes earlier than usual.

8.  Go to the counter to check in and say you have a firearm and/or ammunition.  They will have you open the case for inspection and weigh the ammunition.  You will sign a firearms declaration card acknowledging the firearm is unloaded.

9.  The agent will usually tell you to wait on the side for 15 minutes incase TSA needs you to open the case for inspection.

10.  Upon arriving at your destination, your luggage/case either appear on the baggage carousel as usual, or you have to pick it up at the airlines baggage office.  I recommended you go to the baggage carousel ASAP to ensure someone else doesn't take your firearm.

Note:  Delta Airlines zip ties the luggage/case closed.  Have a way to cut the ties off like a nail clipper or TSA approved trauma shears in your carry on, or knife/scissors in the check in luggage.

Will check that out. Do I need to carry any paper work with me?
Mahalo.

No as they don't check registration.