Re: creating a garden bed.
I would start small; pick a prime spot (good light, the least stumps to be contended with still in the ground, near a source of water and convenient otherwise to you to care for it/enjoy it).
Then depending on what you want to plant (woody things like shrubs and trees or herbaceous things like perennials?) I’d either put the corrugated cardboard down first (with perennials…then make slits to insert the plants) or after planting (if you are putting in a relative few trees and shrubs this may be easier; either way is OK that works for you).
And yes, compost would be a great addition — you can layer it on top of the soil before the cardboard, or if you want to dig holes for your plants work it into the holes…again, I don’t know what you want to plant and how bad the underlying soil is.
Since it has been very dry generally speaking I’d probably water the area first, and I’d definitely moisten the cardboard once it’s in place and then mulch on top and water again. You don’t want cardboard to prevent the plants from getting watered, so leave good openings around them.
One more thing: If there are roots of brambles and other nasty things left still in the ground, I’d unearth them with a pry bar or pick or whatever and not pretend they’ll just disappear because of the cardboard.