Stain or paint ?

We definitely want new hardware. So, that's a given. I originally wanted to do the chalk paint in a light gray, but, my husband wants a dark stain. I have agreed to try to stain it first, I just don't know where to start. Will just simply sanding it first be enough?

Truly staining it would require it be stripped to the wood and stained with the darker stain and finished, probably with polyurethane. It can be done. It is messy work but gives the best result if done well. You want to be sure the doors are solid wood before you do a total refinish.

Gel stain can be applied over the present finish, but then it is really acting as a paint.

For a nice paint job do a good cleaning, light sanding followed by 2 coats of a good bonding primer like Zinsser and 2 coats of a good cabinet paint.

Chalk paint is a possibility for a distressed look.

Yes. You can paint over the stain. If you finish with poly or other high gloss finish, I'd wait for a couple of weeks until it cures. Sometimes another coat will lift what's underneath so let it completely dry. Check a spot by putting your fingernail in the finish In a hidden spot. If it's hard and doesn't indent, it's completely dry. If you make a mark, it's not quite dry. Wait to paint.