My walls were textured plaster so we had someone skim coat them (Spackle and sand) to make them smooth. Right now they are smooth but if you rub your finger in it- it’s dusty.
Can I paint normally (primer then 2 coats color) or do I have to use some special drywall primer before the paint primer?
Your paint job will only be as good as your preparation job. The first thing you need to do is get all the dust off the walls.
First vacuum them, then use a damp (not wet) cloth to wipe them down. Then feel the walls to see if they still feel dusty. Go all the way from ceiling to floor, corner to corner on any surface you'll be painting. Repeat this process until they don't feel dusty any more. If there was a lot of sanding going on, this could take a time or two or three to get all the dust off.
If you paint over dust, it'll show through the final paint coat, probably much more obviously than it does now. Especially since the rest of the wall is texture free. Also, the dust won't adhere to the wall nearly as well as the paint and any dusty areas will be the first areas where paint will just start falling off the wall.
Once you're done, use an appropriate primer for your plaster walls, then use as many color coats as necessary to get good coverage.
