How do I paint kitchen barstools?

You could use polycrylic

Chances are you live in a more humid climate. Each coat needs to dry thoroughly between coats. Can easily take a couple weeks for the paint to cure and harden.

Try sanding the primer and get rid of the sticky. Interior latex primer is best. Polyurethane is a good topcoat over your paint. It give protection from wear and spills.

I painted my kitchen cabinets with milk paint from General Finishes. I used the topcoat from General Finishes. The paint is made for refinishing wood cabinets, etc., and the topcoat is easy to wipe clean. Maybe this would work for your bar stools? You can see the paint in my post: https://momhomeguide.com/my-painted-kitchen-cabinets-a-year-later/

Wow, just saw all of your helpful advice. Thank you so much. Think I just need to get this job done.