Congrats on your purchase! First, you need to find out what type of silver is on the saddle. This will determine how you should clean it. You can find out what you have from the company (you should be able to find contact info for the saddle's brand online, and have the serial number ready).
Sterling and sterling overlay are the higest quality, and are typically very expensive. I highly recommend Haggerty's silver polish. It comes in a blue spray can or disposable wipes, and I get mine from Sergeants' or sstack.com This product was created for use on fine silver, and is gentle enough to use on sterling (which is a very soft metal).
Silver plate is typically a lower quality, but should still shine up pretty well. It often has a protective coating on it, and when that starts to wear off you'll think the silver plating is coming off. Silver plate isn't quite as soft as sterling, so you can use slightly harsher cleaners--Wenol, Simichrome, and even Neverdull. Again, I get these through tack catalogs that carry western show tack.
The lowest quality silver is Inca or alpaca. It's not really silver, but is just silver colored. It will not shine no matter what you do. However, it does tarnish. So, getting the tarnish off and getting the lighter color back is about all you can do. Brasso works great! But be sure you know what metal you're working with, as Brasso is NOT what you want to use on sterling or silver plate.
After you clean up your saddle, you'd be wise to invest in a padded saddle carrier. Not only will it keep your saddle from getting dusty and dirty, but it protects it from darkening in the sunlight and getting any scratches on your silver. Country Supply usually has a black quilted carrier for around $35-$40, and Smith Brother's sometimes runs a sale on matching saddle and bridle carriers.