This idea may be a bit more expensive, but will ensure a good end result. Purchase enough 1/4-inch-thick plasterboard/drywall sheets to cover the existing wall.
Screw or nail the panels into place. Be sure the screws are long enough to fit through the new wall board as well as the existing one.
Place joint tape along the seams, filling in the joints and nail holes with spackling compound. Feather the compound out from the center to eliminate grooves or mounds in the finished product. Allow the compound to dry completely.
Gently rub the dried joint compound to a smooth finish with fine-grit sandpaper. Wipe all loose dust from the wall when finished to provide a clean surface.
Prime the bare wall with drywall primer and paint, or paper the wall.
If the popcorn surface has been painted and does not flake off when you run your hand over it, you might be able to plaster over it. Applying a layer or two of skim coat can give you the smooth surface that you desire. This option is not suggested if the popcorn is loose since it would pull loose, eliminating the possibility of strong adhesion to the walls. Test a small area first.
Lightly clean the surface with soap and water to remove any loose dirt or dust. Allow it to dry thoroughly.
Protect any trim work from the mud or joint compound by applying painter's tape or masking tape along its edges. Remove wall plates and cover any receptacle boxes, and be sure to turn off the electricity.
Fill any cracks or holes with joint putty.
If you did not buy premixed buckets of joint compound, mix the dry joint compound according to the directions on the product.
Start at the top of the wall. Using the putty knife or trowel, apply hot mud or all-purpose joint compound evenly on the wall, smoothing as you go. As it dries, you can knock off small imperfections with the edge of the putty knife.
Allow the compound to dry, and then apply a second coat to fill in any remaining rough spots on the first coat. Let the compound dry.
If the surface is still not smooth, a third layer of compound may be needed.
After the wall is dry, use the fine-grit sandpaper to remove any tool marks and smooth the wall.
Brush off loose dust, and then coat the newly surfaced wall with drywall primer/sealer before adding paint or wallpaper.
If you do not want to scrape the ceiling or wall, you can try sanding. Use a coarse-grit pole sander across the whole surface. The popcorn finish should loosen and fall off easily. Be sure to sand evenly, starting in one corner and moving across the wall.
Go over the surface with a fine-grit sander to eliminate any rough or raised spots. If you have any gouges in the surface, fill them with joint compound, let it dry and then sand.
Wipe the dust from the walls, and prime them with drywall primer. Apply the paint or wallpaper.