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Get shrink-free spackling. Scoop out some spackling from the container with a putty knife. Press the spackling on the knife into one of the nail holes. Use a damp cloth to wipe away any excess spackling around the hole. Let the spackling dry for 2-3 hours. Sand off excess spackling with a fine-grit sandpaper.
Shrink-free spackling won't shrink when it dries in the holes on the trim. Avoid spackling that shrinks or you could end up with dips in the holes you fill in. You can find shrink-free spackling online or at your local hardware store. You can use painter's caulk or water-based wood filler as alternatives. You don't need a lot. You just want enough on the knife to fill in a nail hole. Scoop out the spackle with the end of the knife so it's easier to press into a hole. Start with the edge of the knife on one side of the hole at a 45-degree angle. Scrape the knife across the surface of the hole to the other side. Press firmly down on the knife so the spackling completely fills in the nail hole. Scrape across the surface of the hole two or three times to make sure the spackling is flat. To smooth out the spackling and/or fill the holes, you can also use your finger. Do this immediately after you spackle all the holes in the trim so the spackling doesn't have time to dry. Check back on the spackling after a few hours to see if it's dry. If it's not, let it continue drying. If you notice a dip in the spackling inside the nail hole, apply another coat and wait 2-3 more hours for the new coat of spackling to dry. Lightly brush the sandpaper across the surface of the trim where the hole is until the spackling is flush with the trim.