stain or varnish or...

Staining adds color and is absorbed a bit by the wood. Varnishing is a protective top coat. When you varnish, look for something with UV inhibitors, such as Min Wax Helmsman poly.

Sanding is good for the spot where nothing is left, but you may want to strip if in bad shape, such as cracking, checking, peeling. I'd put three coats of poly varnish on.

The stain isn't gray - the wood is oxidized. The difference between stain and varnish - stain is color only, varnish is a clear coating that protects the wood. For wood doors with that sort of sun exposure, paint is a lot better - varnish has a sort of a "greenhouse" effect on wood - paint will keep sun off wood (in the shade, so to speak). If the cracks are all the way through the wood, you need to apply a flexible caulk or filler before finishing. Sand off the gray stuff.

Unless you sand everything to the same level, you'll have some degree of color variation. After you do your initial sanding pass, wipe the door down with mineral spirits. It will color the wood just a little, giving you an idea of how it will look with finish on it. If you like the look, go ahead and apply the finish. If not, do more sanding.

I recommend a spar varnish or spar polyurethane. "Spar" means the finish is appropriate for use on a boat's spar, which bends a lot, and the finish is flexible. A door that gets direct sun changes size during the day, so a flexible finish is a good thing. Also check for UV resistance.

There is no permanent finish for exterior wood. They all have to be redone regularly. The best advice I've gotten, especially for west-facing doors, is to use a penetrating oil finish instead of a film-forming finish. With the oils, you just paint on another coat every six to twelve months. With film finishes, you have to sand or strip first.

Redoing exterior doors is not simple and to much to go into without seeing it, if its a quality door, I used to spend 50-100 hrs restoring one door and I did maybe 30 of them. You need to strip it, sand it to new wood, and use a Marine finish if you want it to last, go to a real paint store to talk about it and read up, it can be a big job, even get bids from pros.

Aside from any other comments you might consider getting a sun shielding film for the glass in the storm door.