Most hollow-core doors are plywood with veneer, so a light sanding ought to be all you need. The end of my kitchen cabinet, which you saw the corner of, is one piece of veneer ply since I couldn't afford a piece of real wood that large. I did a light sand and stained it with a Minwax stain with a varnish mixed in...turned out fine. Just to be safe, maybe get a small sample from a lumberyard to try; at the worst, you might have to do a coat of sealer such as shellac first, then stain.

Too late to help now but maybe someone else will benefit. I gel stained my Pella fiberglass entry door 6.5 years ago and it looked great! Used Minwax gel stain and their spar varnish, as directed by Pella. However, it did not stand up to weather at all. It chipped and had to be varnished again every 6 months. This Spring it chipped all the way through to the fiberglass and pulled stain with it! Believe me, I did everything properly. My door faces south east and has no roof. But a neighbor whose fiberglass door faces north had the same trouble - he used the same products as I. My neighbor got disgusted and stripped his down to fiberglass and restained it with Sikkens products 3 years ago. No trouble since. Sikkens makes a Door and Window stain that does not require varnish over top. Fortunately for me, just before I started stripping about 36 layers of varnish off my door, it delaminated! Pella brought me a new unfinished slab (under warranty) and I'm using Sikkens to stain it. I've been told the varnish is the trouble. It's like putting a sheet of glass against the door, trapping the heat. Plus it isn't flexible enough to expand and contract. If I put up a storm door to protect it, the problem would've been worse due to trapped heat. Sikkens is specifically made for fiberglass doors. I don't have any vested interests in the company but thought I'd try to help the next guy... Here is a link that might be useful: Sikkens Door and Window stain

No idea on the wood of the cabinets, but I can offer some insight on the staining question -- my husband and I just finished gel staining ours this afternoon! We used GF Java, and contrary to what you're thinking, the stain doesn't look "painted on" unless you...paint it on, and you don't need to strip your cabinets. Our kitchen and bath cabinets started as this (in my opinion) horrible, orangey/honey colored oak. First, we did the bathroom cabinets, which we wanted lighter than the kitchen. To get this look, we applied the stain with a foam brush, and immediately wiped it off with painters rags. We only needed one coat. For the kitchen, we wanted to get them as dark as possible to hide the wood grain, as we plan to go with a busier granite counter top. To get this look, we meticulously painted three thin coats on with a polyester brush. For both techniques, we started by washing the cabinets thoroughly with TSP and warm water. After they dried, we sanded the finish off with medium grit sandpaper (120 sheets and a 150 block, to be precise). The beauty of gel, is that you don't have to have a totally barren surface for it to take. Next, we dusted everything off with terry towels to eliminate debris. Then you just apply the stain using whichever technique you prefer. If you do more than one coat, make sure to very lightly buff the surface with steel wool between each layer of stain. We sealed with two coats of GF polyurethane in a satin finish. It took time, but was actually really easy -- and we love both looks!

If you are talking about real wood think about maintenance. We had a stained mahogany door at our previous house. The sun did a number on it and it had to be redone yearly. It was even under a covered porch but the front of the house faced east and the sun still came under the porch and torched half of the door. Went with a painted door in current house and the front faces west and the door still looks good. It is under a porch as well. As beautiful as wood doors are I would not do one again.