First step is pick up
The Guide to US coins. The
Red Book. That will give you a rough idea of the grade of a particular coin and a sense of its value. Its not that hard to get a rough sense of the coins grade. Can you read the date? Is there wear in Jefferson's hair? Does it still have mint luster? These are questions you can answer yourself.
Hopefully you will find some coins that have some real numismatic value. Mom might have picked up some real gems that have multiplied in value over the years. The next step is to catalogue the coins. Make an inventory, Denomination, Date, Mint, Variation, approximate value.
When you have a sense of the retail value of the collection, then you have to look at options. Local coin dealers will off you wholesale prices (50%-30% off retail). Thats the easiest. Walk in with the coins, walk out with a check.
Go to a coin show and talk to people. Show them your inventory. Determine the best way to sell.
This forum is run by people who will look at your collection and sell it for you.
If you have a lot of value in the collection, you may explore auction houses like Stacks and Heritage. They will take 10-20% plus fees to sell off your collection. However these guys are looking for coins with values over $400, and really want to see $20K collections.
ebay takes a bit of work but you can get near retail value for the coins. Take a picture, make a listing. Start at 99¢. Have envelopes, a scale, and shipping materials ready. If you have good pictures the buyers will determine the grade and value.
Try Coinbug.com. Post a picture and a price and wait to see if you have any buyers.