For the purposes of this quilt, the best fabric you can choose will be 100% cotton. Most craft, fabric and hobby stores have a section simply labeled "quilting cottons". If you're wanting to do a scrap quilt, I recommend digging through remnant bins and hunting around your house for clothes to deconstruct and cut into squares. I did very little shopping for this quilt - the fabrics are a mixture of my grandmother's and mine. :)
If you're wanting to use two colors or do something more spectacular, you'll need to do a little math to figure out how much fabric to buy... here are some figures to get you started!
Number of blocks and squares in a queen size quilt:
Blocks: 48 (10.5 in square sewn size)
Squares: 432 (3.5 in square sewn size)
If you're doing 2 colors, you'll need 216 of each color in the quilt.
Number of squares you can get out of ONE YARD of fabric (this is assuming you've trimmed off the selvedge ends and are left with fabric 40 inches wide):
90 squares (since your fabric will be 36x40 inches)
I figure you guys should be able to figure the rest out based on those - but if you need any additional help, don't be afraid to ask! :D