Painting walls after taking down wallpaper
This really depends on how the old wallpaper was installed, what type it was, if the walls were primed before and so on.

Best case scenario (happens 10-15% of the time I have to redo walls) is that your walls were painted/primed, they didn't glob on glue, the glue has broken down due to age and you simply pull off wallpaper and paint.

Good case scenario is that after you pull off wallpaper you are left with some felt/paper or some glue residue.  Here I suggest using super hot water and vinegar and a grout sponge (rough side).  If it is really bad you can go through with fine drywall sandpaper (black mesh).

Bad case scenario is wall paper has to be thoroughly wetted and scraped off using a wall paper scraper.  Then you have to fix issues with drywall mud and sand that down, prime, and paint.

Worst case scenario is wall paper is stuck on the wall and the walls haven't been primed.  Wall paper tears off drywall top layer no matter what you do.  At that point its a test of will.  Will you tear down all the drywall (the sane/faster way but also "harder") or will you diligently scrape off every single inch, do tons of touch up which equates to skimming mud on the whole wall...