Old wooden chairs can be painted or refinished to natural wood and stained or sealed. Metal outdoor chairs can also be repainted.

[edit] Steps

Cushioning either has to be removed or completely masked off with heavy paper and tape.
The procedures necessary are relative to the desired end result:
Metal Chairs: Outdoor metal chairs are generally painted with corrosion-resistant paint such as Rust-Oleum, usually oil based, but all sorts of epoxy, polyurethane, alkyd and other primers are available today. Chairs can be painted as they are or after some degree of rust has been removed. They can be painted with a brush, however over metal, the best finish is achieved with spray. Rust can be partially or completely sanded away with electric vibrator sander, and coated with a rust sealer such as Zinc Chromate or Red Oxide Primer which may be quite toxic.
Wooden Chairs: You can simply paint over the existing finish. You can use oil or water base paint if this last is water base enamel. Latex is unsuitable. Oil base generally gives a better finish and feel over wood and is more durable.
Sand the rough spots with a vibrator sander or by hand. A disk sander will leave circular grooves. Start with a rough grade of sandpaper like 100 grit and move to finer sandpaper to finish (use anything up to 320 grit). Files may be used for hard to reach places and you can make different shapes by wrapping tools with sandpaper.
For a complete strip and restoration of the natural grain, a paint stripper might be used. This is a paint dissolver and so may be quite toxic. Rubber gloves and plastic eye goggles and gas mask would be necessary. Water base paint strippers are also available but am unaware of how effective they are. Strippers are generally banned from land disposal so you will have to check on disposal method.
The stripped wood can be painted, stained or finished clear. Varnish adds an amber tone to wood and has a nice feel, while polyurethane adds no color but might tend to have an overly grip feel.

[edit] Tips

You can use tack (sticky) cloth to remove all dust before painting.
Old wicker chairs can be painted with primer for seal, and sprayed for best finish.
You can create your own colored stains by adding just a little tint or paint to stain and test before using. The same is true of wood filler. However the stain should be the same base (oil/water) as the paint or fill.

[edit] Warnings

Do NOT inhale paint and stripper fumes. Make sure to do all painting and stripping outdoors or in a well-ventilated room, to avoid intoxication from the paint stripper.