What do I put down on this flooring? Do I need something different in the tray versus in the nest boxes? Do I leave the tray empty so I can scrape the poop off everyday? If I put the same material down in the whole coop, what's easiest? Sand? Pine flakes?
First and foremost,

welcome to the BYC community!
I see that
Tractor Supply sells a variety of prefab coops. Which one did you get? Sounds like you should be OK with any of the coops if you have 3 chickens.
Since you have a pull out tray for cleaning, I would recommend pine shavings based on the shallow depth of the pull out trays I have seen on prefabs. You could also use pine shavings in the nest box. Another option that I have currently had success with is using shredded paper I make at home for free. I used shredded paper for deep bedding in the coop and also in the nest boxes. I would say that shredded paper works just as well as pine shavings, but pine shavings do look better. On the other hand, paper shreds are free, so it's hard to beat that aspect.
I have never owned a prefab coop with a pull out tray for cleaning. I suspect that you will be cleaning out the coop a lot as the chicks get older and their poo gets bigger. Although pine shavings are relatively inexpensive, paper shreds are free (for me) and if you end up using them for only a few days, then you will save money using paper shreds.
FYI, I bought a couple of paper shredders for less than $5 each at our local thrift store. There is some satisfaction used paid bills as chicken coop litter, if you know what I mean. I was shredding all our bills anyway, so I just started shredding all our junk mail, newspapers, and light cardboard food boxes. It all makes good litter.
Although I live on a lake, and can get all the free sand I could ever use, I am not a fan of using sand at all. In my experience, sand needs to be cleaned out all the time because it will get wet and smelly really fast. It is also much heavier than pine shavings or paper shreds, and that might be a consideration for your pull out cleaning tray. Also, I would not put sand in the nest boxes.
If your prefab is new, I would suggest you take a good look at that pull out tray and see if you can reinforce it now before it breaks. From what I have heard, that is about the first issue chicken owners experience with their prefab coops.
I hope to hear more from you and see some pictures of your birds. Everyone loves pics. Show off your birds.