The easiest and cheapest way to remove wallpaper is to use warm water and a sponge or rag to dampen the paper. Wait several minutes for the water to permeate the paper and breakdown the glue. Then, simply find a crease in the paper or a loose and pull the paper off in sheets. This sounds too simple and many times it is. Occasionally you will have paper that is this easy to remove, however, most times you will need to use some or all of the following steps in order to fully remove the wallpaper. Remember, if the wall was primed/sized before the wallpaper was hung and as long as the paper was hung correctly then it should come off fairly easily. Also, there are different types of wallpaper and some are easier to peel off than others. Provided this first step does not work for you try the following steps.
Many times you will need to use a solution to aid in breaking down the glue of the wallpaper. I prefer to use a product called DIFF and mix that with the warm water (follow the directions on the bottle), however, many people use liquid fabric softener or other products which are also effective. Once you have mixed the solution you may want to “score” the wallpaper. Scoring the wallpaper is simply putting tiny holes in the surface of the paper so that the solution can penetrate to the glue backing easier. This can be accomplished in a couple of ways. You can either use a scoring tool or I have heard of some people using steel wool or sandpaper. I prefer the scoring tool which can be found at most home improvement or paint stores. Simply rub the spiky tool over the surface of the paper (getting every square inch is not important) then wet the wall using your sponge or rag soaked in the solution. Allow to sit for 5-10 minutes and begin peeling the paper using a wide blade putty knife or fingers (or whatever works for you). Always try to start at a joint or a torn edge of the paper. Strip off as much of the paper as possible then wet down the remaining paper and glue and peel that off. Done properly this step usually completes the job for me, whether I am dealing with wallpaper or border. However, should this step still not work than try step number 3.
STEAM! You can rent a wallpaper steamer, purchase one at a home improvement store, or try to use the steam function on your iron or garment steamer (the last two are not as effective). I still prefer to score the wallpaper first then using the steamer (follow the directions for how close to the wall you need the steamer head to be) slowly cover small sections at a time of the wall, then immediately (while that section of the wall is wet) scrape off the wallpaper. Continue this for all of the walls. Even after the paper is removed in this manner, you will still need to try a solution (mentioned in step 2) or, at a minimum use warm water and a rag, to remove the glue still adhered to the wall.
You thought you were done! Almost, but there is one essential step before you can repaint the wall. Prime the wall with a primer meant for applying after removing wallpaper. This is in case you missed any wallpaper glue on the wall, the primer will allow the paint to stick. If you simply apply paint over wallpaper glue residue you are in for a heap of problems. After you prime you can topcoat with the paint of your choice. Good luck!