Your first quilt is just beautiful! ;) You have done a wonderful job and I think it looks MUCH better than my first quilt did! Lol*
As for how much backing fabric you want to buy - here is the way I do it. I layer my batting giving myself about an inch and a half (but I eyeball it) of additional batting around all four edges of my quilt. Then I layer that on top of my backing and I want to see about an inch sticking out around all four edges of my batting. So you see all three layers when quilting. You can get away with a little less, but give yourself more batting and backing than your top, your quilt will shrink up a bit when you quilt it so you'll need that extra. Then you'll trim and square it up after you've finished quilting. You may have to piece your backing to fit or you can purchase backing fabric that usually comes off the bolt in 60" wide yardage.
I use all kinds of batting. A low loft cotton is the "in" thing these days, but years ago people didn't dream of using anything less than the highest loft polyester batting they could find! It's personal preference. Cotton is a little heavier than polyester, and I like the look myself because it is reminiscent of antique quilts. Unless you are making a King size quilt though cotton doesn't usually get so heavy that it would be uncomfortable.
I often use Hobbs Heirloom 80/20 batting (80 percent cotton and 20 percent polyester) which I buy at Jo Ann's in a bag. They are already precut so you can get a crib, twin, queen, or king size cut. Hobbs is really nice because they treat each side of the batting so that there is no bearding, even with wear. It's probably my favorite batting but a little more pricey so I save it for my best projects.
More often I choose Warm and White or Warm and Natural sold in 90" wide cuts off the bolt at Jo Ann's. I believe it goes for something like $10.99 a yard. You should be able to get a two yard cut of that and have plenty for your quilt. I believe W&W is the better choice for machine quilting while W&N is a great choice for hand quilting. But I have used both for both purposes and been happy with the results. Also this goes on sale a lot at Jo-Ann's and I believe they even have a sale coming up soon on batting. This does brand does beard a little bit so I would use a topstich or comparably fine needle when you're quilting.
I think sewing x's diagonally through your squares would be a great way to go - or even a free motion meander. I understand you're still new to quilting but don't be afraid of free motion. A meander is REALLY simple to do and I think is even easier than trying to get perfect lines accross your quilt.