Stain then polyurethane or paint them. Pictures of the doors and homes interior may help you get other ideas.

Real cherry? I'd skip the stain and all the headaches involved with that process (eg blotching) and just leave them natural. I've never seen a stain on cherry that improved the appearance, it's one of the most beautiful woods in the world as is. Note that cherry darkens significantly over time with exposure to light and air.

Sand to 220 grit with a random orbit sander, then three coats of varnish with a light sanding between coats. If you use polyurethane note that there is tremendous variation among products in terms of what the final appearance looks like. I tested over 20 brands to pick the one I wanted for our woodwork. Ended up with the highest VOC oil-based product from Last-n-Last. You may like something different. I thought the Minwax products were among the worst in terms of having a pastic-y appearance. I used gloss for the undercoats, then semi-gloss for the topcoat.

Others here will argue for non-urethane varnishes. I've never had many of the issues they cite (chipping, for example) but it's worth considering.

I find the water-based urethanes quite ugly myself, but if you choose one you should add a step to your prep process-- at the final sanding grit go over the wood with a damp rag to dampen but not soak the wood. You'll find your previously smooth surface is now fuzzy. Using fresh sandpaper at the final grit knock off the fuzz with a very light sanding. I often do this by hand rather than with a sander. This will ensure that the surface is as smooth as possible when you put the water based finish on. (this step will also improve the surface for an oil-based finish, it just isn't completely necessary)

My wife sold furniture for years. Customers would order cherry furniture and complain that it did not look like "cherry" when delivered. She would explain that with a few months, it would take on that deeper, redder coloration of cherry due to natural oxidation and exposure to light. It was actually a problem for the room settings in the store which had decorative items sitting on them. She would have to go around every couple days and move those items to a new location, or risk that a mark would develope where it was sitting.

If the cherry is merely oiled or covered with a protective lacquer or urethane finish, the darkening will still occur until equalibrium is reached.

One reason stains are applied is that they not only give initial color, but they stabilize that wood from darkening naturally.