This weekend, I'm all set to go handgun shopping since I've wrapped up my FSC. Is there anything I can bring along to actually make the purchase happen?
If you want to be safe.
Driver's License, preferably California.
Copy of your Birth Certificate
Passport
Copy of Hunting License if you have one, or any type of membership to a range.
Copy of Safety Cert (CA Hunting License requires that online, so it's good to use that)
DMV registration to Address buying firearm from.
There can still be snags. As an example, I use my p.o. box for the current address. However, you can have a P.O. box on your DL, but some stuff will not ship to P.O. Box, such as Hazmat.
By having your passport can work around that, and/or hunting/safety license/tag. CA will put the hunting license and the first tag on on receipt, and each game thereafter.
If you want to minimize the chance of having to go back until your firearm gets out of jail, the above items are good to have and may save you a trip back. Either way you will need to go back to pick up the firearm, and it wouldn't hurt to talk to the store that will DROS it for you. My local FFL will use a copy of DMV Reg, passport, DL, Hunting License/Safety.
The above is how my FFL handles long guns. Handguns may be limited to how many you can purchase in a given 30 day window.