Do I use mud or spackle after removing a popcorn ceiling?
If your working some large areas or going over taped seams then your choice should be the mud. It goes on smooth and is very easy to sand. Spackle is great for small holes but not really good for this much work. 
As far as how many times do you need to apply the mud that depends on what you're trying to fix. If you're covering a new taped seam then plan on three times: with a 4", 6" and 12" knife. Torn paper and nicks one or two coats. Mud does shrink so touch ups might be necessary. Good luck. 
Use a good quality paint with the primer added to it.