The most important thing I do to a fixture is work you can't see. Rewiring antique fixtures is the heart of restoration, and when it's done right it's invisible: the fixture looks exactly like its old self and runs like a new one.
The hazard you can't see
Old wiring is the problem restoration solves. The cloth-covered wire inside an eighty- or ninety-year-old fixture was fine when it was made, and it's a hazard now. The insulation dries out, cracks, and flakes off the conductor, and a fixture that looks flawless can be dangerous behind the canopy. The old sockets wear out. Grounding, if there's any, often isn't up to a modern standard. None of that shows from across the room. So I pull the old wiring out and run new, rated for present use, with sound sockets and proper grounding where the fixture allows it. What I keep is everything that makes the piece what it is: the body, the shade, the finish, the lines.
Now the part where I stop. I'm not going to publish a step-by-step for doing this yourself, and it's not gatekeeping. Line voltage is unforgiving, the failure modes hide where you can't see them, and a fixture wired wrong passes every look-it-over check right up until the day it doesn't. This is work for someone who does it daily, with the piece in hand. The honest thing I can offer isn't instructions; it's a fixture that's already right, tested and ready to hang.
Salvager's Note: When a listing says "rewired to safety," it should mean new wire, sound sockets, and grounding where the fixture allows, all invisible from the floor. Ask what was actually done to the wiring; if a seller can't say, treat the word as decoration.