What are the steps used to prep/paint/seal acrylics onto wood that gives you a durable painting?

Because the stools are shiny, you need to sand them before you do anything. You don't have to sand them down to the bare wood, you just need to take the shine off. Wipe down with a damp cloth after sanding to remove any residue. Make sure the surface is dry. Then you can use primer to start with.

Once the primer is dry you can paint anything you like on it. Once the painting is finished, you will apply clear polyurethane to seal it.

Edit: If you don't sand the shine off, nothing is going to stick to the surface.

Sand the top of the stools with a 120 grit sandpaper. You want to ensure the shine is dulled. This also scuffs the surface to allow adhesion of the primer. Ensure to clean the dust before proceeding.

Apply an acrylic primer. Let thoroughly dry before lightly sanding with 220 grit sandpaper. Clean dust.

Apply 2nd coat of primer if after inspection you can see bare wood. Sanding after dry followed by cleaning.

Start applying the team logos starting with the background layered colors, applying the foreground colors after each coat has thoroughly dried.

Once the team logo is completed and has dried thoroughly, apply 2-4 coats of clear polyurethane to protect the surface.

You will find patience is a must to achieve professional results since most of you time will be spent on preparation and waiting for the previous coat to dry.

I have painted on many stools. First sand the stools, clean the dust off, prime them with house paint primer (I often use Kilz 2), then paint them with acrylic latex house paint (I usually use an enamel paint). The enamel paint if allowed to dry for about 30 days is very durable.

If they already have varnish on them, I would consider the surface already prepped. Go ahead and paint and get some sealant or varnish to protect it. I would epoxy the top because I'm paranoid, but that's kind of overkill. I think spray varnish would do fine.