Although some wallpaper removal techniques work faster than others, there’s no “quick” method. If you try to rush through it, you'll just gouge the drywall and end up spending more time overall having to repair all these spots and clean sanding dust.
First, if the wallpaper surface seems slick, determine if it’s the vinyl-coated type. Using a sponge, dampen a small spot, and wait about 30 minutes. Try to scrape the area with a wallpaper scraper. If it isn’t loose at all, then it probably is vinyl coated and it will resist water.
By your description of its appearance, let’s assume that it’s vinyl coated. The trick to stripping it as quickly as possible is to break the surface in as many spots as possible. This allows the water to penetrate and soften the old glue so that you can peel off the wallpaper.
If you’re careful, you can score the wallpaper surface with the edge of a scraper. Score many times in a crisscross pattern. Just break the surface. Don’t score it too deeply, or you'll get into the drywall paper coating, and this will cause problems.
Another alternative is to buy a special wallpaper-piercing tool at your paint or hardware store. It has spiked wheels inside that pierce the wallpaper surface when you roll it over the wall. Spending a little extra time rolling it to create many holes will save stripping time overall.
Now you have prepared the wallpaper, and you’re ready to begin the stripping process. Plain warm water is effective, but using the enzyme-based additives will dissolve the old glue much faster. DIF enzyme stripper, by Zinsser, is one of the best products to use.
Spray the enzyme solution on the wall with an old window cleaner spray bottle, or sponge it on with a large sponge. Have plenty of ventilation. Although the enzyme solution isn’t harmful, it makes some people cough. Be patient and give it about 30 minutes to dissolve the glue.
Spray another section of the wall, and let it soak while you scrape the first section. Use a standard wallpaper scraper. Holding it at about a 15-degree angle seems to remove the wallpaper well without excessive damage to the drywall underneath. Don’t worry about a few nicks in the drywall.
Once the wallpaper is removed, you'll have to scrub the wall to remove all the residual glue. If you don’t do this, it will bleed through the paint in a year or two and look terrible. Use a large grout sponge and the enzyme solution. Wash it down with soap and water when you’re done.
Now, unless you’re the Michael Jordan of wallpaper stripping, you have probably damaged the drywall in some spots. To fix these gouge spots, first paint them with white shellac or other stain sealer. This blocks the moisture from the drywall joint compound from penetrating the drywall.
When the shellac is dry, fill the gouges with a thin layer of the drywall joint compound. Use several layers, and sand between coats. When completely sanded and painted, the repairs will be invisible.