Whatever you choose, do the work outside so your home won't be filled with fumes.
What finish to use? That's easy, whatever matches the rest of the doors in the house and the wood trim in the room!

You could use different stain colors for different stories of the house, or even for different rooms especially if the room's functions are completely different. But don't mix-and-match colors in a single room.
For a formal, vintage, or masculine appearance use a darker or deeper color. To make a room lighter, 'sunnier' or more informal go with a lighter color. Using more coats of stain and/or letting the stain sit longer before wiping off will give a deeper tone. For the most 'natural' look, so the woodwork
looks like pine when you're finished, use one of the pine stain colors - you'll need some kind of color stain to bring out the grain and give the door some 'warmth'.
Whatever you choose try it on a bit of scrap wood first. Preferable a scrap of pine wood (most lumber is pine), because stains and finishes look
significantly different depending on the wood. You can buy small cans of stain very inexpensively - buy several that interest you and try them all.
I have stripped and refinished several old doors that had been painted over, much of the house's woodwork, and many new carpentry items as well. The easiest, fastest, cheapest, and most durable process is to use a Minwax stain and a polyurethane finish. The stain fumes are the most noxious-smelling, but that's locked in once the poly is applied. There are other brands of stain that are 'supposed' to be better, but don't over-stress about it; just use whatever brand has the color you like best.
There are some water-based polyurithane products. The finished product seems almost as good, the dry time is faster, and the fumes are more tolerable.
Here are some of the Minwax colors I've used:
For the home's library, all of the woodwork (all pine) was finished in "red oak". A deep color that really brings out the wood grain without being dreary or too dark.
On the home's third floor I used a new color, English Chestnut. I love it! A deep beautiful color that gives an aged or vintage look.
In the entry room Wife used Golden Pecan in order to match the existing color on the main staircase. The golden pecan is almost like the pine stains, but with slight tint.
Did a pine bookshelf in jacobean. Very dark, darker and more black-ish than expected. I wouldn't use it for a door or a room's trim. Maybe if going for an exotic look?
Simple steps to follow:
1 - Sand the door as fine as possible.
2 - Apply first coat of stain. Rag, cheap brush, whatever.
3 - Allow first coat to soak in and stand for desired time. More time = deeper, warmer color.
4 - Wipe off excess.
5 - Let stand for however long (according to instructions on can), longer for deeper color
6 - Steps 2, 3, 4, and 5 again for second coat of stain if desired
7 - Let stain dry completely before adding polyurethane. Time listed on stain can.
8 - Apply first coat of polyurethane.
9 - Let stand for instructed amount of time
10 - lightly sand or 'scuff' the new poly layer.
11 - Apply second coat of poly.
Obviously do all of this with the door laying flat so the stain and poly doesn't run or streak.
And definitely do the work yourself. Paying someone an hourly rate to stain woodwork?!?