Dealers and collectors are an option for taking 78s off your hands. But they are also the toughest to sell to. Dealers and collectors often buy 78s through the mail, so that means you'll have to provide a lot of detail about each record's condition to get a price you want. But the price you get also depends on the what is in the collection, as Tim Gracyk says in his article "The Value of old 78s" on his website. The majority of 78s still around today are from the 1940s and are mainly big band or classical music. There's nothing wrong with these styles of music, but they are so common for 1940s record collections that when grandma buys the farm and you're forced to sell her records, most serious collectors aren't going to care about her big band or classical music collection because everyone has a big band or classical music record collection. If you are serious about getting a good price for your 78s, Gracyk's website will be your best friend; he offers a lot of tips for determining the value of 78s, selling 78s, information for beginners and advanced collectors as he offers his insight as a collector.
Antique stores and used record stores are an option for you to sell your 78s, but word of caution, you won't get much money for the collection. Most used record stores are going to offer you a fraction of what you can probably get for the record on an Internet auction site or from a dealer. You might have a gem somewhere in the collection, and most record store employees or store owners will simply give you a flat fee per record--such as $1 per record, or worse, 50 cents per record. While that pricing example is more common for LPs (33 1/3 RPM records from the 1960s to 1980s), it could just as easily happen at your basic record store with an employee who doesn't know the value of the records. To sell your 78s in an antique store, oftentimes you'll need to sell them to an antique dealer who pays you for the records and then makes his money back when the records sell. Unless the antique dealer is a record collector, who knows if you're getting a fair price? The dealer or store might pay you one price and then turn around and get more money for the records.
Believe it or not, the Internet and especially eBay might become your best friend if you're trying to sell a 78 and want to get a good price for it. By selling it on a site such as eBay, you can see what other records like yours are going for as well as the demand for each record. You can evaluate the record; when you make the listing on eBay, set a price for each record based on what other like records are getting. If it's a hot item, or a good price, you could get enough bidders to drive the selling price up. The other benefit to a site such as eBay is that you have more options. You can sell individual records, you can sell the entire collection or you can break up the collection and sell it in lots. It's best, however, to sell your gems as individuals so they don't get lost in a collection or in the lots.
Think garage sale, as far as records are concerned. Records typically hit the auction block or are included in an estate sale when someone dies; that usually means the family wants to get rid of items quickly. Without an assessment of what is in the record collection, the family could be selling off some real gems for a good price to the buyer. As Tim Gracyk says, if Lawrence Welk is in the collection, chances are the collection isn't worth much based on 1) the listening tastes of its owner and 2) the fact that Welk records are a dime a dozen. The other problem with estate sales and auctions is that the buyers who come to these sales are looking for a good deal; when the family is pressed to sell something, oftentimes they are going to get less than what the items are worth.