How you choose to finish (or not finish) each project is totally up to you. You will want to keep in mind the use of the item, however. For example, a trivet to be used habitually for hot casserole dishes is essentially a sacrificial piece between the countertop and the dish. In time and with extensive use, any trivet may discolor and, in some cases, show burn marks. That is to be expected.
For extra protection, you can add a clear heat-resistant spray or brush-on lacquer. You can also use a clear water-resistant spray or brush-on lacquer when using a trivet as a centerpiece on a table with a flowerpot that might get wet, for example. For trivets designed for outdoor use, apply a clear weather-resistant spray or urethane for UV protection. Make sure to allow any final finish to dry completely before you use the piece.
lf you are placing a trivet or coaster on a tablecloth or a wooden surface, attaching felt or cork spacers/feet to the bottom is advisable. This will help prevent oil from bleeding into the tablecloth if there is no additional lacquer coating, and it also protects the surface from potential scratches. Felt or cork spacers can be purchased at many hardware, dollar, and craft stores. Make sure that the spacers are thick enough to allow a space between the trivet and tabletop surface.
Felt spacers/feet like these add an extra layer of protection to your home's surfaces.
Painting with acrylic paints is an easy way to jazz up a design.
For coaster finishing, l like to apply a wood sealer first, then two or three coats of a spray varnish or hand-painted varnish such as Varathanc or urethane.
When you plan to use a trivet or plaque for decoration and want to add color, acrylic paints are a good choice. You can buy small jars or plastic bottles from any art supply store. Many dollar-type stores carry them as well. Seal the wood first if you wish, or simply apply the paint directly to the raw wood. After painting is all done, I recommend coating the entire piece with a clear lacquer, such as satin or semi-gloss. lf you do not like using spray, use a protective coat of Varathane or urethane. Your local hardware stores will have both, so experiment with finishes to find the ones that work best for you.