Some depends on the primer but since you have it all ready, I would prime it all to get the same base to work with, only using a QUALITY primer
Zinnser Cover Stain or 123 would work as would Fresh Start

I'm with christophern, hit all 5 surfaces with a quality primer for the best outcome. On ceilings: it seems like a hard job to primer and paint a ceiling, but it's not that big a deal. It does help to wear a hat and put a drop cloth on the floor. Using a roller on an extension is great arm exercise and a bit of Zen besides.

I'd prime everything with Smart Prime by Zinsser.

Thanks for the good advice. I'd prefer not to use an oil-based or alkyd primer if I can help it. Will Smart Prime, 123 or Fresh Start seal any residual wallpaper paste that might be left on the walls?

Probably not but Gardz will

I was going to ask you if all paste was removed but didn't want to assume you didn't.

If you are unsure, then GARDZ is your friend.

If you are unsure, then GARDZ is your friend.

ditto that

Thanks for the unanimous advice about GARDZ. Do I need to put a different primer on top or do I put two coats of paint on top of the GARDZ?