Process can vary depending on airport, airline, airline employee you talk to or day of the week. I've gone to IAH 2 weeks apart, United both times and had to do two completely different things because the employee I got was a moron.
Generally you have your gun unloaded in a locked hardside case, non TSA locks, (although I've had discussions at the airport about that before). Go to the full service desk (although that can also vary), let them know you have a firearm to declare. They'll get an orange card, maybe they'll fill it out for you, maybe they'll let you fill it out and sign. Put that inside the case, close it up and lock it.
At Houston airports IAH and HOU they just throw it on the belt and will call you over the intercom to open it if TSA wants to inspect it. Other airports they might walk you over to the TSA area and x-ray it in front of you, others always make you open it to inspect it, some others make you stand in a spot until they give you the ok to go on. It varies a lot.
When you go to pick up your luggage sometimes it goes to the airlines baggage claim office (keep your tag), or oversize baggage if it's a big case, and sometimes it just comes out on the belt.
My best advice is know the laws, airline rules, but if an airline agent or TSA agent is sure they're right, and it doesn't hurt anything just let them do what they want. More than once I've had an employee tell me my bag wasn't going on the plane unless I did what they wanted to do.