How do I prepare walls for wallpaper after removing wallpaper?
It depends on the wallpaper. Some "splits" to leave a lining paper behind when the face comes off. If this is sound, you could decorate over it.

If the remaining is uneven, yes, take it off.

To prep the walls, make sure that they are smooth, no loose plaster, etc. Sometimes you'll see walls washed down with sugar soap to remove left over glue or other mess. Fill or caulk any uneven bits, then paper over.