Lampshades are expensive and fragile. It is worth it to take the extra time and effort to pack them properly.
1. Do not pack anything else with lampshades, unless you want to spend money buying new lampshades.
2. For packing normal lampshades, begin by getting large boxes, clean packing paper, and bubble wrap. Put bubble wrap on the bottom of the box. Line the sides of the box with clean packing paper. Put the lampshade top-side-down in the box. Crumple paper in the empty spaces within the box and inside the shade. If you are going to nest smaller shades within the larger shades, then use bubble wrap as a buffer. Top the shade with bubble wrap.
3. For packing glass lampshades, real care is required, since it is often the case that much of the value of the lamp is in the shade. Get a box that is somewhat larger than the shade, but not overly large. Place a pillow in the bottom of the box. Tape packing paper tightly on all of the glass. Then wrap the shade with bubble wrap. Place the wrapped shade in the box on the pillow. Take crumpled paper and fill in the corner spaces. Then place another pillow on top of the glass shade and fill-in top corner spaces with crumpled paper.