In some ways, a gun is just a piece of personal property and crossing state lines is no big deal. In other ways, it may be highly regulated and when you cross state lines, the rules may change a LOT.
It makes me crazy to read some of the complacent opinions I saw above. I have LEOSA and a collection of state carry licenses, like FL and UT because they are broadly accepted. I have mentioned in other settings our RV project. I plan to carry smaller guns with smaller mags so I don't get sideways of the unconstitutional mag limit laws. Most cops (and lawyers) are ill-taught about a firearm as evidence in a shooting, so when I leave Washington I tend to carry lower value firearms that are easily replaced (Sig 239, M&P Shield, the like). Because so many states have silly laws about ARs. my new Marlin .357 will be the primary long gun, but it will have an RDS, big flashlight and a sling, all set up as much like an AR as possible. Far less likely to become a political prisoner that way.
If you are not sure of the legality of what you are doing, research it, and print it. That means the statutes, not a summary by someone else. If you have a firearm in the car, keep your nose clean about getting stopped. Sorry to say that the response above about an MC endorsement is on the far side of irresponsible. Take a few hours and get legal. If you are not properly endorsed as required by your home state, no cop in his right mind will let you drive away. They should impound/tow/arrest, and that will lead to a search. You will have an awful time winning a suppression hearing and if you do, that will still be an arrest and a big expense. If you know certain states and cities are stupid places to be, don't go there. Have your family gathering in PA, not NJ or MD for example.
If stopped carrying and lawful, comply with state law about notice, but don't use dumb language (not: "I have a gun"; the better way to express it is "I am lawfully armed ..."). Depending on where you are, the cop may say s/he does not care as long as you don't go for it. Other places, they may take it and unload it, and be a real pain. First answer: don't do stupid stuff and get stopped. 5, maybe 10 over is ok. More than that, you are taking a bad chance. What may be tolerated in state A might not be in state B.
NEVER consent to a search. If they get a warrant, you are going to have to comply. If they don't, they lose. You have nothing to lose if they ask why you are not cooperating and say "I don't have time for a real search". A well done search as if they are looking for drug couriers could easily take several hours for the search and then reassembly of your stuff. Have a good idea if you are taking a road that has a lot of drug smugglers (I55, 57, 65, 95 for example.) Don't tempt fate. Years ago I went to a Pat Rogers class in SC. Washington plates on I95 will get a lot of curiosity, and cops are supposed to be curious. This was especially true since they were actually running an interdiction class. I was followed and really expected to get stopped. As a man on the path to old and broken, I had a lot of prescribed stuff. In some states you do need a 'script for syringes; your state law will not be a defense. I had a bunch of gear in my trunk, too. Mack (rottie) was in the back seat, and I had "work dog/stay back" stickers in my tinted back windows. Most real cops like dogs better than people, so they understood the car being locked up and running.
Don't like my position? Fine. Remember I do this stuff for a living. Do not come crying to us if you screw this up.