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If you get inexpensive unfinished wood, it can be pretty rough, and you may need as coarse a sandpaper as 100-grit. But for most pieces you can start sanding with medium-fine sandpaper, perhaps 150-grit (Image 1), to smooth out any nicks and scratches the furniture has picked up along the way. Sand in the direction of the grain. Finish with fine sandpaper, say 220-grit, to achieve a smooth surface. Once you've settled on sandpaper, get out the sanding block (Image 2). It's contoured nicely to fit your hand and also to keep your fingertips away from the wood. Note that you don't need the power of a belt sander for furniture -- save that for floors.
Remove the dust created by sanding. If you're going to use an oil-based paint, remove the dust particles with a tack cloth, which is basically cheesecloth with varnish added to it for picking up dust. If you're going to use latex paint, you can pick up the dust with a damp cloth or sponge.
The two decisions go hand in hand. Latex paint, which is water-based, is now as durable as oil-based and doesn't have any fumes, which means you can work indoors. It's also environmentally friendly. If you're using latex paint, you'll need a nylon-bristle brush. For best results, buy a medium-quality brush for around $6 -- there's no need to buy the $20 model.
Paint sticks better to primer than it does to raw wood, so put a coat of primer on first. Primer normally comes in white, but you can have the paint store tint it to the same shade as your paint. Dip your brush into the paint, covering only about a half-inch of the bristles. If you oversaturate the brush with paint, you risk leaving brushstrokes on the furniture. Brush the primer on in the direction of the wood's grain, working it in with short strokes as you go. Prime all the surfaces.
Let the primer dry and then sand it lightly, remembering to wipe off the dust from sanding afterward. Apply your second coat of paint, also painting in the direction of the grain of the wood.