I wish I could offer a lot of practical advice, but that would mean I'd have to do it myself!
Because I'm still working on getting my sewing room to be a better place for me and because I'm trying to get some extra steps in each day, I've moved my cutting area to my dining room table. That helps because I try to keep everything where I can just pick up the cutting mat and move it. But it is absolutely amazing how many rulers and other accoutrements add up, measuring tape, pins and safety pins and on and on...
I keep my fabric to the left on the back of a chair. I sit in the center. On the right is another chair, on the seat is my box I put my little scraps in, I sent those to a crumb quilter (you may have a designation of your own scrap box). The strips I'm keeping for my cutting go on the back of the chair, along with any 6.5" strips I cut for my own scrap box (I no longer save anything smaller than 6.5").
I make a project box for each project. Sometimes that is a washtub from the dollar store, sometimes it is a cardboard box. Sort of depends on the size of the project and the finished block size. When I start cutting subunits, they get counted and into the box they go. Again the scraps go into the box for the crumb quilter. Typically the project box is on top of the crumb box so the crumbs stay on the cutting board until subunits are completed.
When the large pieces of fabric are cut down into strips, they go back into the sewing room. I keep them together (typically in an open washtub so I can see them) until my project is ended so in case I need to go back they are handy, after that they go into their designated boxes.
Or at least that is how it is supposed to work...
Edit/PS: It helps to keep a trashcan handy for the little bits of fluff and trim. I deliberately don't so I get up and walk another 10 feet to the trashcan every few trims.