Some people might argue that wood furniture should never be touched, but it really depends on the condition of the piece and the look you'd like it to have. I am not opposed to painting wood furniture but believe that should be a last option. If you can find a way to make a piece work in it's original finish, great! If the color just doesn't work with anything you own consider first staining it to keep the wood grain look, or paint it if you want a completely different color.
When refinishing a piece of wood furniture the first step would be to analyze it (maybe start sanding it down) and decide on your end goal! Next strip any current paint and sand the whole thing down. I would recommend using a power rotary sander for best results. This step could be very simple or take quite a while depending on the condition of the piece. If it is pretty rough or has detailing you want to remove, start with sandpaper of 5 or 10 grit. Then to finish it off use a 100 grit paper to get it perfectly smooth and on the delicate parts. Use extra care on edges and detailed trim that you do not want to remove, depending on how intricate you may want to hand sand these parts.
Before you start to paint or stain, you want to wipe down the wood and clean off all the dust from sanding. Make sure you have an open clean work area, preferably outside or in a basement with doors and windows open. Lay down a tarp, some cardboard or paper bags etc. anything to protect your floor. Have an extra spare rag to wipe off your hands and excess paint.
When staining you don't want to use a paintbrush, instead use an old rag or old sock to avoid brush strokes. Stain the sides first making sure each stroke goes all the way from one end to the other, following the wood grain, to avoid dripping and uneven color. Try to do the top surface last as typically that is the most dominant face and you want it to look best. Allow the stain to dry at least 60 mins before applying a second coat, if you do more than two coats allow even more time between the latter coats as it may start to get sticky. The more coats you do the darker the color will appear but also the less the wood grain will show through. Adding a top sealer coat is a good way to finish it off and keep the wood protected, use a rag or sock for this step as well. Once finished allow the piece to sit outside a few more hours and completely dry before moving it or placing anything on it.
When painting wood the process is pretty much the same as staining, except you do use a paintbrush. Again make sure each stroke goes from one end to the other, limit lifting your brush to avoid brush strokes or any drip marks. Paint in the same direction of the wood grain. Start with the sides, and finish with the top surfaces for painting as well. For small edges or detailing you may want to go back over them with a small paint brush. Depending on the color you should only need one or two coats of paint, again the more coats the darker it will be but the less the wood will show. Paint fumes smell very strong so giving it a few hours outside to dry will help eliminate bringing the smell inside.
One of my first wood furniture projects was refinishing a small dresser I had found at a yard sale. It was all beat up and an unattractive light color (no hesitation to leave it as is) so I sanded it down first which brought out a beautiful wood grain and then decided to stain it with a pale yellow finish. The pale yellow color did need three coats. It was by no means perfect but I embraced the flaws and hand sanded it afterwards to give it a more distressed look!
My next wood piece I refinished was a wood desk that I got for free from a friend, he had already started to paint it so there was really no question of leaving it how it was. The narrow legs needed very delicate sanding as not to wear them away but the rest required lots of sanding to remove all the nicks and bumps. I decided to go with a dramatic black paint, that looked great after just one coat. The inside was already a pretty natural wood color so I left that as is which looks even better now with the contrast of the black! I did use caution when painting the door so it wouldn't get painted shut. I also removed the drawer and painted that separately so it wouldn't get stuck shut. As you can see in the third photo, I did not lay anything down and ruined my deck (oops, learn from my mistakes) luckily it isn't much and my rug covers it.
My most recent project was refinishing my kitchen table. We got the table for free from Brian's dad when we moved into our place. It was originally a tuscan wood color, but pretty beat up. The wood color really did not match our light wood cabinets or match anything in the kitchen so I always kept a white tablecloth on it. So once I had already bought black paint for my desk there was no reason not to paint the table too. Although after sanding it down I did consider leaving it that light unfinished wood color, but knew that black would be a more dramatic contrast in the kitchen like I wanted. It turned out great and only needed one coat of paint, you can still see the wood grain which I love.