I got some kind of green felt at Walmart. I just put the chess pawns on it & used a light coating of Elmers glue on the pawn edge bottoms. Worked perfect. Trimmed the felt the next day with a sharp pair of scissors.
billiard felts for chess pieces
In that case just put a light coating of Elmers glue on the plastic bottom. Stick it on the felt & let dry overnight. Trim with a sharp pair of scissors the next day & you should be good to go. The thin coat of Elmers is the secret... too much on the bottom & it will soak through the felt.
On wooden pieces, Elmer's school glue in a very thin layer is perfect for attaching felt (provided the wood where you are attaching the felt is sanded bare), because wood and wool felt are both porous natural fibers, which is what Elmer's excels at. I've used it myself for reattaching portions of felt to wooden pieces that were lifting (I used Elmer's wood glue because that's what I had onhand at the time; pretty much the same thing as Elmer's school glue), and it has been fine for 15 years, and is still fine.
For plastic pieces, I wouldn't trust Elmer's. Put a dab of Elmer's on a plastic piece and let it dry; I guarantee you could easily scrape it completely off with your fingernail, as if it were never there .
I'd suggest a high quality spray adhesive, such as 3M Super 77. Follow the instructions on the can and be sure to mask off the rest of each piece with e.g. masking tape and newspaper so that the glue only gets sprayed on the bottom.
Anyone know a cheap source for refurbishing/retrofitting chess pieces? Do you have to buy it by the yard and use spray adhesive? Advice appreciated.