Shows are for dogs - get out and DRIVE.
That being said, what you really need to do before you start driving is go over it with a VERY fine tooth comb for road-worthiness. You cannot assume anything with a purchased car that you didn't build, no matter what the P.O. tells you. That wiring looks scary as hell IMO. What is the reason for the rat's nest going on there? Could be ok, could be an electrical fire waiting to happen. If it's bad then toss a new wiring harness in it. With the kits available today it's really not that big a deal. A good weekend project at best, and it's also a good time to convert to 12V if it hasn't been done yet.
Even more importantly - BRAKES. Get that thing up in the air and scrutinize every line and hose. Look for rusty or kinked lines, cracked or dryrotted hoses, leaky fittings, etc. Pull off all the wheels and drums. Inspect the linings and drum surfaces. Make sure the wheel cylinders are functioning properly. Check the diameter of the drums and make sure they are within tolerance. If they are, have them turned if they are grooved. Pop open the master cylinder and check out the fluid. Should be nice and clear, not murky looking. Pull the hubs. Clean out the bearings & races and repack them with fresh grease. Adjust the brakes properly. While it's in the air check all the suspension & steering components. Check ball joints, tie rods, pitman arm, drag link, A-arm bushings, springs, etc. Check all fluid levels - engine, transmission and differential. Check the tires for uneven wear, foreign objects or broken belts. Find a local alignment shop that actually knows what a '54 Chevy IS, and take it in. Have them put it on the rack and set the alignment.
This is just a short checklist, but things you should most certainly do before driving the car any distance at all. I know it's cool to hop in a old car and go for a drive, but it's not like the daily driver you're used to. It's a bit of work, but well worth it. If it was easy everyone would do it. We want to make sure you and your wife can enjoy this hobby for many years to come. Good luck!
P.S. I spent many a gunnery mission at ol' Graf. I was stationed in Budingen (now closed) with 1-1 Cav. from '92-'94. Good times over there. Thanks for your service. [ddev