National Parks/National Wildlife Refuges
Section 512, Credit CARD Act of 2009 (54 USC § 104906 and Title 36 Chpt. 1 Part 2 § 2.4) changed federal law to make park firearm regulations reflect state law. Typically, the only restrictions on firearm carry are state and local laws. If it is legal elsewhere in the state, it is legal in the park and sections that conflict with state law regarding carrying and possessing firearms (but not shooting bans) do not apply. For example, carrying a firearm or possessing loaded firearms in a vehicle is prohibited (unless one has a special park permit, usually for hunting) in National Park Service lands (National Parks, Monuments, etc.). However, this does not apply if the state allows open and/or concealed carry or loaded/unloaded firearms in vehicles.
The park buildings (visitor centers, ranger stations, offices, fee collection buildings, and maintenance facilities, etc.) are still federal facilities and off-limits to firearms. These places are marked with signs at all public entrances. Discharge of firearms, except when lawfully hunting, is generally prohibited. View the NPS pamphlet here. The same applies for National Wildlife Refuges, 16 USC § 1a–7b and Chapter 50 Chapter 1 Part 27 § 27.42.
National Forest lands do not ban firearms. However, as in National Parks, ranger stations and visitors centers are federal facilities and are off-limits to firearms.
Airports/Aircraft 49 CFR 1540.111 and 49 USC § 46505
Under federal law (local laws may differ), firearms are prohibited beyond the TSA checkpoints. You cannot attempt to enter a TSA checkpoint while armed, even by mistake.