A coin dealer would be a good place to sell it.
You can sell it at coin shops, flea markets or to collectors you may know. There are other options of selling but unless it is uncirculated there would not be much money left for you.
From 1797? They would be very valuable. However, with coins, it is all about condition and rarity. You would need to consult an expert to determine the value of the coin. 1792 is the first year the US coined money.
Ebay or a coin dealer. Don't expect much for it, unless the coin is in exceptional condition, you might not even be able to sell it for more than two cents. It is a very common coin.
Any reputable coin dealer should be able to value the coin for you. Look under "Coin Dealers" in your local Yellow Pages.See the link below.
Find a coin show in your area, take the coin, and get offers from several different dealers.
At any coin shop. Beware though, you aren't going to get much for it, especially if your coin is worn. It isn't a rare coin.
You could sell them at a coin store, eBay, or bring them to a coin show. And find out what there worth.
Well, an uncirculated one would sell for whatever 1 ounce of silver is selling for, today about $27. However you say with proof, I'm not sure what that means, if you mean it has some sort of certificate of authenticity or it is a proof coin. A proof coin will have mirror like fields (parts of the coin with no design) and an uncirculated coin will have shiny, but not mirror like fields. A proof coin is more expensive and would sell for around $70.
The best place, and most convenient for you, would be to go to a coin shop and offer it to a dealer. You will not make as much as if you were to sell it online, but when you sell it yourself you are accepting the risk of losing the coin in the mail, having an unsatisfied buyer, BUT you will more money for it.
These sell regularly at coin shows for a dollar or less.
Assuming you have a British coin, not much. 1948 shillings contain no silver and are quite common. Such a coin would sell for about a dime in US money, or about 5-7 pence in UK pounds.