Here's some brief info on gun ownership.
The age requirement you asked about, it's 18 for shotguns/rifles and 21 for handguns. Everyone I know has gone through a background check, but there is the gun show loophole, where private sellers don't need to do background checks, which is something I'm against. The background check itself is very loose, they do a criminal check and use the honor system for other stuff (do you abuse alcohol? do you have a history of mental illness? etc). Again, something I'm against and wish there was a way to properly screen people.
For Texas, there is no waiting period, while California has a 10-day wait and Washington has a 5-day. Others vary, I'm not sure what they are. When I bought my handgun it was a 10-15 minute process. State laws also vary on carrying in public. Texas you need to take a class and the weapon must be concealed, while some places, like Colorado, allow open carry and without a permit. Like this guy:
In Texas, and I'm sure others, there are limits to where you can conceal carry. Any business that has 51% of it's income coming from alcohol sales prohibits carrying. As do hospitals, government buildings and places that specifically say no guns. Generally schools don't allow it either, though there are some Universities that allow it, and of the places that do allow concealed carry, no gun related incidents have occurred.