Staining wood furniture is generally a necessity after you strip and prepare it. If you don't, the piece of furniture will quite often look uneven in color or kind of blotchy. An exception to this is a clear finish over light color wood that you want to have a natural appearance.

You can get products that are a combined pigment and finish that dry quickly (other than the pigmented oil finishes), but they will generally give a muddy opaque finish. There are vinyl or aniline stains, and many more that are quick to dry, but are usually very difficult to use or just plain frustrating, because they dry before you can get the effect that you want. It's just like most things in this life, if you want fast, then you have to give up some other quality.
There are pigmented oil finishes, that generally give good color. Watco Danish Oil is a good example.
My favorite is a good oil stain, they give you lots of control. Always pick a color no darker than you want the furniture to be. If the furniture is too light you can always apply more stain and let it penetrate longer, but it is a lot of work and sometimes the result isn't what you want if you have to lighten the furniture. If the color you want is in between the colors available, you can mix oil stains to get your desired color.
Oil stain can be applied with a brush, cloth, roller, your bare hand, or by any other means as long as you get it on evenly. After applying, wipe the excess off and let the furniture dry overnight. It is very important that the stain is completely dry before applying a clear finish.
 

Somewhere along the way someone has come up with the notion that you should seal the wood before you stain. If you want to save yourself some disappointment, don't seal before you stain. If you seal the wood before you stain, then there is nothing for the stain to soak into and you end up with a smeary unstained appearance.
Sealing after staining is a very good idea, some stains will bleed through a finish giving an undesirable appearance and it is a good idea to seal them in. Besides, using a sanding sealer, you can sand it with a very fine sandpaper to give a very good, very smooth surface to apply the final finish to.

Be sure that you use a sanding sealer compatible with your stain and with the final finish product.

You have gone through the difficult part, stripping and preparing your project, now it's time for the final step, applying a finish.
There are many products available for finishes, some very hard and some quite soft. Some will give more protection than others, but some pieces of furniture don't get very hard use, so will do fine with a less protective finish.

Staining wood furniture is generally a necessity after you strip and prepare it. If you don't stain the piece it will quite often look uneven in color or kind of blotchy. An exception to this is a clear finish over light color wood that you want to have a natural appearance.

Your furniture does a boogey woogey all day long. The only thing is that it's drummer is very slow moving so you can't see it happening. As temperatures change, the whole piece expands with heat and contracts with less heat.

Exposed bare wood is very susceptible to moisture changes. If you have electric baseboard or forced air heating it's very dry.

Every time someone takes a bath or shower, even with venting, or when you fix meals there is a teriffic uprise in humidity.

The exposed bare wood expands, then the air dries from the heating system, so there is constant expansion and contraction, but the other side with a finish sealing it isn't as susceptible to the ups and downs of humidity all day.

The warp and wane of the bare wood puts stress on the finish and joints on furniture. Natural finishes, lacquer, shellac, oils, etc. are flexible and can withstand the constant stretching, shrinking, and twisting. That's part of the reason for joints loosening and why hard inflexible polyurethane will crack along a joint line.

If you have read anywhere else on this site, you probably have figured out that polyurethane is not one of my favorite finishes to use on furniture. Although there have been improvements in polyurethane, it still is a hard plastic coating that doesn't have much give. If you are working on an antique and you use sandpaper and polyurethane, you no longer have a provable antique and the antique value will drop through the floor.

There are many different kinds of oil finishes, some are easy to use and some are very time consuming.

Linseed Oil was the choice of the old timers.
I put the boiled after the linseed oil to draw attention to it rather than to indicate that there will be other types covered. The other type is raw linseed oil, it will never dry, but will become gummy and sticky, so be sure to get BOILED linseed oil if you decide on that kind of finish.

Linseed oil gives a fantastic finish, but you need a year to apply it. The general rule of thumb for a linseed oil finish is once an hour for a day, once a day for a week, once a week for a month and once a month for a year. The finish was usually French polished once a year after that. That's a lot of work.

The other oils are much easier to use. Danish Oil and Tung Oil are two popular oil finishes. They are very easy to use and come in clear and in pigmented colors. Just follow the directions on the container, but double the number or coats they recommend. They aren't as durable as some other finishes, but are very easy to repair. If you get a light scratch in the finish, just grab a cloth and apply another coat of the finish and generally it's all taken care of.