ATF Form 5320.20 for the Class III stuff. IIRC Suppressors do not need ATF approval to be moved interstate, but you might as well so you don't wind up in a conundrum with an LEO ignorant of the law.
Non Class III stuff just pack it and move it.
Do be careful with Kalifornia. I moved up here from Tempe in 2003 and Kalifornia proclaims to not give a shit about the '86 GOPA. If something happened and they took everything away, you could go to court, fight them and maybe win, but thats a lot of time and money. Might consider the scenic drive up to Vegas then Reno rather than I-5 or 99, still have a little corner of CA to cut across, but only 100-miles or so. Hate LA so would always take I-10 out to Quartzsite then zig-zag across the dessert a bit, cross the border at Parker, to Needles turn left onto I-40 to Bakersfield (<-shares the shithole-of-the-world-award with El Paso) then turn right up 99 to Sacremento... much nicer drive than I-10 to I-5.