Walls and Ceilings - Untexturing my ceilings

I'm in the process of renovating my 1873/1910 farmhouse and decided to remove the texturing from the ceilings. (ceiling texture was poorly done sometime in the 1970's) I've scrapped the living room ceiling to remove the "popcorn" and it came off rather easily because the ceiling was never primed or painted, just texture over the drywall - what a mess!! Anyway, now I'm not sure how to proceed. I think I should probably skim coat the entire ceiling to even out the inconsistencies before I paint but:
1. do I need to do something before I skim coat it (there is a fair amount of drywall dust left behind after scraping)
2. although I've done quite a bit of wall drywalling, I have very little experience with ceilings and I've never had need to skim coat. Any tips on skim coating ceilings?
Thank you.

Clean the dust off the ceiling first. If there is any exposed gypsum, prime with Zinsser Gardz before skim coating.

It also helps to thin the joint compound a little. No need to apply a thick coat of j/c, just a thin coat to even out any irregularities...... and if you find an area that needs more attention - it is no big deal to add another coat of mud to that area - even if it doesn't show up until after you apply the primer, just remember to reprime any areas you fix.

Thank you both for such fast replies. Just one more question - what would be the best way to clean the dust off the ceiling?

I usually use a broom, if you are concerned about the dust getting elsewhere you could use a shop vac.

A damp rag/sponge can also do a good job of removing the dust but it might rewet and remove some of the joint compound also.

Damp rag is ok if you're careful not to impart too much moisture to the ceiling. I'd stick with the broom.