Elmer's glue (not washable kind)
Polyurethane spray (I used the glossy kind)
Roll of adhesive cork board (optional)
A little patience
1. Wash and dry tiles. Cut the paper to the right size and shape. Water down the glue to somewhere between heavy whipping cream and milk consistency. Apply the glue-water to the tile.
2. Put the paper on top of the glue and apply more water-glue. Wait for it to almost dry and then apply another coat. I think I did 5 coats, but that may have been excessive.
3. Admire your soon-to-be coasters.
4. After the last coat of glue-water has dried completely spray polyurethane on them. I did 4-5 coats of this also. The tiles will be sticky (and your room will be stinky) even after several hours. I was really worried, but the stickiness went away after about a day.
5. Cut squares out of the adhesive cork board and stick them to the backs of these beauties.
I adore these. I saved one for myself and gave away the rest as presents. (Mine is the green cat one.) I've had it since December. I put 3 cups of hot mugs of tea (sequentially, not at the same time) on my coaster daily and it is yet to show any signs of abuse. I wipe it down every once in a while. That's about it.
Here are some links to other people making coasters: