I fly all the time with firearms. I've recently flown in and out of California twice with hand guns. It's painless.
All you need is a lockable suitcase and a lock. You will have to check it. When you get to the baggage check, let the agent know you are traveling with a firearm and you have to have TSA take a look at it. They will point you in their direction. I've brought several back wrapped in towels but a hard case wouldn't hurt. Generally TSA will have you open the suitcase and show them the gun. You will fill out a tag , then they will have you lock it, then they will tag it and take the bag. You keep the key.
If you take ammo with you, most airlines have limits. Most limit you to 11 lbs. Check the airline you are flying with.
When you land, you will have to pick it up at a secured bag claim. It won't be on the regular carousel.
As for California law, they require new residents to file and ownership report.
If you are going to claim to be a traveling Texan, you may choose to ignore it. In the two times I have been there with guns, I have never been asked about it. If they do, tell them about the form and let them know you have 60 days to file it.