I'm moving from California to South Dakota. Is it legal to transport my shotguns in my car?

There are no overly restricting states between CA and ND so there shouldn't be any problems. The Safe Passage clause of the Firearm Owners Protection Act applies if several conditions are met when transporting firearms. 1) The firearm is not accessible to the driver or occupants (locked case in the trunk). 2) and the ammunition is unloaded and kept separate from the firearm.

Good rule of thumb regardless of where you're going is to put the guns in the trunk cased with a lock and place the ammo in a different compartment!

I have travcelled all over the western Us and have never had any problem transporting firearms. I drive a pick-up so carry my long guns in the back seat, under a cover. Unloaded and no ammo in reach.
Enjoy SD,, this is gun-friendly country.

I don't believe there's any restriction on it, but just to avoid stressful situations if you're stopped for any reason, it might be a good idea to break each of them down before putting them in the car. No one will feel threatened by a disassembled shotgun.

Wait, you currently live in Cali and own a gun? Stay right there while I call the cops. All kidding aside, check here: https://www.nraila.org/gun-laws/state-gun-laws/. A lot of information.

Or, just call each state's Highway Patrol and ask how to do it lawfully.

Only after you leave Kalifonia

Do not even consider making a side trip to the Mexican border with weapons in your vehicle. Warmest Regards, Sandy :)

http://www.dailymail.co.uk/news/article-2816530/Mexico-judge-orders-release-Marine.html