You can paint over wallpaper, if the wallpaper is still adhering well. You would need to first go over that with an oil-based primer to seal the paper. If you were to paint the room with water-based paint, then the paper would start coming off because the glue on the paper would be activated, that's why you must first seal with oil-based primer.
I would suggest just tearing off the paper. Depending on the paper, you can just spray with warm water and let sit for a minute or two, then pull it off. For some you would need to wet down, then pull top vinyl paper off, then spray again on the paper left on wall to activate the glue and then it just pulls off reall easy. I wouldn't waste your money on wallpaper remover. Recently I just took down four wallpapered and bordered walls and it was not a problem at all just using the spray bottle with water.
As far as paint, if you want a really great paint, then go with Sherwin-Williams or Benjamin Moore. Those costs a little more though than your usual box store paint. But I am currently using Valspar Premium from Lowe's and it's a good paint, I love it. Stay away from Behr from HD, not so so good, in my opinion. I would suggest Valspar though because either way you will need to do two coats if you are rolling it on, no matter what you use. And two coats of Valspar works just fine.
