Is it legal to bring (or allow someone else to bring) a firearm on a private flight?
This is a tricky question, because it depends greatly on the laws of the locations you're flying to / from.  Here's a few of the things you have to take into consideration.
Transporting is generally ok

According to 18 U.S.C. 926A:

Notwithstanding any other provision of any law or any rule or
regulation of a State or any political subdivision thereof, any person
who is not otherwise prohibited by this chapter from transporting,
shipping, or receiving a firearm shall be entitled to transport a
firearm for any lawful purpose from any place where he may lawfully
possess and carry such firearm to any other place where he may
lawfully possess and carry such firearm if, during such transportation
the firearm is unloaded, and neither the firearm nor any ammunition
being transported is readily accessible or is directly accessible from
the passenger compartment of such transporting vehicle: Provided, That
in the case of a vehicle without a compartment separate from the
driver's compartment the firearm or ammunition shall be contained in a
locked container other than the glove compartment or console.

926A says nothing about the method of transport.  Most often it applies to motor vehicles, but it can just as easily apply to aircraft.  Therefore as long as you're ok to carry at the departure airport and the destination airport, federal law protects you in between.
Also, remember that 926A only protects you if the firearm is unloaded and locked / inaccessible.
Ok, but can I bring my gun unloaded onto airport property, right?

Not in a sterile area
You can't take a firearm into the sterile area of an airport.  49 CFR 1540.5 defines a sterile area as:

Sterile area means a portion of an airport defined in the airport
security program that provides passengers access to boarding aircraft
and to which the access generally is controlled by TSA, or by an
aircraft operator under part 1544 of this chapter or a foreign air
carrier under part 1546 of this chapter, through the screening of
persons and property.

So, if the terminal and the GA ramp are two separate entities, you're probably ok.  However, I've flown to some smaller airports where the GA ramp and the TSA terminal are right next to each other.  In these cases, you need to be careful that you don't get into the sterile area with your firearm.
Maybe not anywhere on the airport
Local law might prohibit the possession of a firearm anywhere on airport property, or the airport itself might ban firearms on their premises.  The only way to know is to check the specific airport you intend to fly out of.
What if I want to carry loaded?

The first thing to remember if you carry loaded is that 18 USC 926A does not protect you.  In that case, you're exercising the privileges of either open or concealed carry according to the laws of the departure and destination states.
Open carry
Some states prohibit open carry.  Others allow open carry when unloaded.  Others allow open carry when loaded.  Some require a permit.  Some don't.  Some only require a permit for handguns but not long guns.  If you're going to open carry (perhaps a hunting rifle on a sling), make sure that the local area allows this.
Concealed carry
Some locations prohibit concealed carry outright (to the best of my knowledge, you still can't carry concealed in Chicago, for example).  Other states allow concealed carry with a permit from either that state or a state which they have reciprocity with.  So, for example, a Mississippi permit would be honored in Texas, but not in Nebraska.  Others allow concealed carry without a permit at all (that is, the 2nd Amendment is your permit).  First, be sure that your concealed permit has reciprocity with the destination state.
Even if you think you're open carrying, watch out.  In some states, the act of being in a vehicle, no matter how visibly your firearm is displayed, makes it concealed carry.  If you don't have a concealed carry permit, you may be in violation.
Is there anything else waiting out there to bite me?

Oh yes.
For example, Rhode Island.

A Rhode Island carry permit is generally required to
transport a handgun in a vehicle. This requirement does not apply,
however, to the holder of an out-of-state carry permit who intends to
travel uninterrupted through the state.

Is a fuel stop an &quot;interruption&quot;?
Or New Jersey.

[New Jersey] State law generally prohibits any person from knowingly possessing or
transporting any rifle or shotgun without first having obtained a New
Jersey firearms purchaser identification card (FPIC),or from
possessing or transporting any handgun without first having obtained a
New Jersey handgun carry permit.

Or Chicago.

Chicago, for example, bans handguns declared “unsafe” by the
Superintendent of the Department of Police, so-called “assault
weapons,” machine guns and other firearms regulated under the National
Firearms Act, “laser sight accessories,” firearm silencers or
“mufflers,” “high-capacity” (i.e., more than 12-round) magazines, and
“metal piercing bullets” (which include ammunition with non-lead
projectiles).  All these items are subject to seizure and forfeiture.
In some cases, the local ordinances have no explicit exceptions to
these bans for travelers merely passing through the jurisdiction

Or New York City.

Possession of a handgun within New York City requires a New York City
handgun license or a special permit from the city police commissioner
validating a state license within the city. Even New York state
licenses are generally not valid within New York City unless a
specific exemption applies, such as when the New York City police
commissioner has issued a special permit to the licensee or “the
firearms covered by such license are being transported by the licensee
in a locked container and the trip through the city of New York is
continuous and uninterrupted.”


Look, just tell me what to do.
Fine.

Carry your firearms unloaded, locked, and inaccessible.
Make as few intermediate stops as possible
Know and comply with all local laws both at your departure and your destination
Stay out of sterile areas while armed on airport property