You are at least 18 and preferably 21 ?
18 is the legal age for possessing a rifle in most states. 21 is legal in all.
As long as you are old enough, you should be fine, The only states that are pricks about firearms and transportation of long guns are not in your path, NY, Mass, MD, NJ Ill, and Cal are the sticklers and even those supposedly can not stop you for it if you are just passing thru. but they can make life difficult.
Do you have some sort of bill of sale/letter of gifting/ or the like ? that would help, but most states should let you pass just fine. If your grandma is still alive have her right a letter saying she is giving the rifle from your grandfathers estate to you. Or have your parents do the same. That gives you legal ownership and clears the way for you to have the right to protest should something go wrong.
A zipped case should be enough, if you put one of those little luggage locks on the case that would be better but should not be necessary. Make sure its empty, any magazines (excluding a tube magazine if it has one) should be placed elsewhere and keep the ammo elsewhere too.
I routinely travel from Minnesota to ohio, missouri and alabama and florida with a gun in my possession and that never seems to be an issue.
Remember, Cops do not have xray vision, and unless you give them a reason to go looking for it, they will never see it or find it.