This little recipe from a jewelry friend is an inexpensive and quick way to clean any sterling silver (Tiffany's silver isn't any different than other sterling silver- it's just from Tiffany's)...
I don't know about the pearl part though...cleaning pearls should be done by a professional jeweler in my opinion.
A great way to clean your jewelry is to get an aluminum pan, like the disposable ones. Add really hot water, not boiling, but very hot. Add 2 teaspoons of baking soda and 1 tsp. of salt. Stir a bit. Add jewelry and let soak for 10min. Rinse and dry thoroughly. Viola’ just like new. This will remove tarnish, and also is safe for all jewelry components including swarovski’s and pearls.
I don't know about the pearl part though...cleaning pearls should be done by a professional jeweler in my opinion.