To remove sticker goo on unpainted spoons, I use a cotton ball and Isopropyl rubbing alcohol. Sometimes I try to remove the sticker itself by holding the spoon under a hair dryer for about a minute, then peel the sticker off.
The way I get my spoons to shine (especially the silver plated ones) is by lining a baking dish with aluminum foil, placing the spoons on the foil, cover the spoons with baking soda, and then I add very hot (but not boiling) water; then let them soak for about five minutes. Do not rub them with baking soda, this will scratch them. Then make sure to rinse them off.