I think the most likely cause is the mechanism being gummed up with a mixture of oil, rust and dirt. Thorough cleaning of chain, sprockets, chainrings, deraillers and cables with something like WD-40 or kerosene. Then chainlube like white lightning, a small amount of light oil on the deraillers and cables.
You chain and sprockets may need replacing, but I dont think they would be the cause for jumping between gears, only of skipping in one gear.