There are a number of ways to make a wine cork trivet, such as gluing the corks to a surface or each other; or connecting them using wire or thread. Whatever technique you select for making a wine cork trivet, the first step will always be collecting the corks.
To collect corks, ask your friends for their discarded corks, go to a restaurant that serves wine to ask for corks, or drink a whole lot of wine on your own and save the corks.
Next, select the desired appearance of your trivet. You can frame it, bind it with ribbon, or leave the edges natural. You should also decide whether to mount the corks standing on end or laying down, and determine whether you want to cut the corks in half or smaller to reduce the width of the trivet. Gluing your corks to a metal or wood surface can provide added stability, as can framing it, but securing the corks to one another can provide an aesthetically pleasing appearance.
If you would like to use a backing, use hot glue or an epoxy to secure the whole or cut corks to the surface, ensuring the edges of the corks are flush with one another. If you would like to forgo the backing, use a drill or awl to make holes in the cork and thread twine or wire through the holes to connect them securely.