Painting over textured wallpaper is not difficult to do. Depending on the type of texture, it can actually be easier than painting smooth wallpaper, because the texture can disguise the seams. The method is basically the same no matter what type of wallpaper you plan to paint, and with a few basic tips it can come out looking professionally done.
Check the walls for loose seams or wallpaper coming loose in the corners. If you can't paste it back down with seam paste, gently pull it off. Remove all nails and hangers. Take off the switchplate covers and anything else that might get in the way of painting.
Caulk all the edges. If the adjacent surfaces are wood, use a brown or tan caulking and wipe away every bit of excess caulking material possible. If the adjacent surfaces are painted white (like trim and ceilings), use white caulking instead. There are usually some slight gaps where wallpaper meets the ceiling and trim, and these will really show up after you've painted if left unfilled. Let the caulking dry overnight.
Prepare the room for painting. Use blue painter's masking tape to mask off ceiling and trim unless you have a very steady hand for straight lines. In any case, the baseboards and chair rail, if any, need to be masked. If you are also painting the trim, do it first and then mask it off so you get nice crisp lines. It is much easier to paint trim first, let it dry and then mask it, rather than trying to do it last.
Prime the walls with a shellac or fast-drying oil-based primer. Using a water-based primer or painting directly over unprimed wallpaper is asking for trouble. The water in the paint or primer will lift and loosen seams. If you are planning on using a strong colour for your paint, have the paint store tint the primer as close as possible to the finished colour.
Do any wall or seam repair and cover-up after you have primed. Check for nail holes. If the seams are really noticeable, use joint compound or caulking to disguise them. The method you use to disguise seams will depend on the type of texture in the wallpaper. You may need to get creative with your filling and spackling. Applying spackle to nail holes with your finger might work out better than using a putty knife, to avoid flat spots on the wall. Spot prime all repairs.
Choose a flat paint. This is the best option because the less gloss a paint has, the less it will show irregularities on the wall. If washability is a concern, look for one of the new scrubbable flat or matt paints. These are usually not available at big-box home stores. Go to a real paint store and ask the staff for their best scrubbable flat paint.
Count on doing two coats of paint, even if the primer is tinted to a similar colour. Two coats of paint will look better and be more durable. Always do the cut in (brushing) before rolling to minimise brush marks. Wait at least two hours after the final coat to remove the tape and replace switch-plate covers and anything else you have taken down from the walls.
When pulling off masking tape, pull it slowly backward instead of ripping it straight up.
Use plenty of ventilation when priming, and don't let birds or small pets breathe the fumes.