Run your hand over a sunny surface of the shed. If any chalky residue comes off, the shed must be power-washed or scrubbed free of chalk.
Use a power washer or scrub brush to remove chalk and dirt. If there's mildew, wash the shed with a detergent containing mildewcide. Scrape loose or flaking paint, then rinse the shed until the water runs clear. Let it dry completely.
Thin exterior oil-based primer with about 1 pint of mineral spirits per gallon (a single gallon of primer should be sufficient for an average storage shed). Protect the ground around the base of the shed and apply the primer with a roller and brush. Let it dry per label directions.
Caulk holes and seams with a good grade of exterior paintable silicone caulking. If you wish, you can use automotive body filler to repair dents and large gouges. Caulking does not need to be primed before painting, but repaired dents should be primed again with the oil primer.
Paint the shed with two coats of 100 percent acrylic (water-based) exterior paint. Avoid painting in very hot sunlight, or if it's extremely humid or threatening to rain.