Take stock of what you have. Make a list of each record you own, as well as the condition of the packaging and the record itself. Even if you have many old records in great condition, much of your potential sale relies on demand. Try prioritizing your list, starting with what you think might be the most valuable records.
Set aside records that were premiers of specific pieces, records conducted by notable conductors like Leonard Bernstein or the composer of the piece being premiered, and records recorded by especially notable orchestras or performers, like Glenn Gould. Determine the condition of each record by comparing your record collection with the guidelines from a record appraising authority, like Prex.com.
There are several routes you can take to sell classical LPs. Consult a record shop online, getting a quote of what they would pay for your records. This won't always get you the best deal, since shops are interested in selling these records back for a profit. Even if you have a rare record in good condition, they might not pay much if they don't think it has resale value. Going to a record shop in person can help, especially if they specialize in old records or classical music in particular. In person, you might be able to haggle or trade records, rather than just selling them all for a lump sum.
Online auction sites, like Ebay, could help you sell your records also. This way, you can list each record individually or sell them in small collections to maximize your potential profits. However, you will have to ship each record, so make sure you charge the buyer an adequate shipping price. Craigslist is another online resource for selling LPs. Craigslist acts like an online classifieds section for your local area. You can list your collection in a music category, and you might not have to pay for shipping because it is likely that the buyer lives near you.