Lightly sand the entire piece of furniture to make the colors duller and scratched. Use a fine-grit sandpaper so that you only distress the finish, not the furniture itself.
Sand harder on specific areas, like the edges and flat top of a table, or the seat and arms of a chair.
Gently knock on the wood with a hammer to create dings and imperfections in the wood. Use the hammer carefully on the edges, where the wood can crack. Of course, depending on how weathered you want the furniture to look, cracked wood may be what you want.
Scratch the paint with a wire brush for uneven weathering, especially along the edges.