Mom left artwork, some valuable, some not so valuable, how do I get it appraised and sold?

To get the most $ your going to need to DIY this.
If your mom kept records of buyers / clients, I’d contact old buyers via a snail mailed letter & perhaps email follow up if you can find their email addresses. Write maybe 10 - 20 to see if they would be interested in buying the collection or perhaps just the lithographs and posters. They or their family will remember the gallery and appreciated her discerning eye.

If there’s records, there will be price lists. You can google each artists and look at recent auction sales..... but I’m tired to even think about this. If it were me & there were old price lists I’d across the board add 20% and be done with pricing. Unless you know there’s something truly valuable. 

Another thought... Have you thought about Etsy? What you essentially have is a shop. It would be perfect for an Etsy site. If you or someone in the family or a friend of yours could open an Etsy, I’d bet you can clear all within a year. The paperweights and books could be in groups so that would be less per item listing cost. These can be shipped out USPO via flat rate with the provided for free USPO boxes. The litho & posters if those are mounted have to go speciality shipping boxes - ULine has these and their catalog is a fab resource to determine what packaging for shipping will cost. If flats, they can get archival paper sandwiched an into tube mailer that uspo also has for free but I’ve found you have to ask for those from the postmaster. 

Storage is very critical. My MIL had a gallery type of shop in Santa Fe and in New Orleans. Lots of bad storage foxing issues with prints & books. Do a gimlet eye once over at everything in daylight. Segregate anything with foxing asap. Those you may not be able to sell as the costs to get conservation done may exceed value. If your needing time to figure out what to do, I’d encourage you to put each item (books, smaller prints) that can fit into their own freezer grade ziploc type of bag. In “theory” they should be archival grade (U line has these too), but freezer grade zippie work just a well. I have a lot of ephemera & other smallwares that use for work and I get freezer grade 2 & 5 gallon ones at restaurant supply stores to store stuff in for a fraction of a ZipLoc branded ones. They safely store stuff till practically forever.

Etsy can seem overwhelming. If it’s not your bent, I’d look to find an Etsy shop owner in your area that can run your shop for a %. Could be a great site if you have old fotos of your mom, the shop or events from her era to put in. 

I'm in a similar process now for some estate assets: Need to inventory, find someone to make repairs and clean up, decide whether to store off site or onsite, consider the insurance value and whether or not theft would be a covered item in a vacant house (big issue!), have them evaluated and decide whether to try to sell myself in an area in which I have no expertise, or hire someone to sell for me. I do know that there's a market for these items though; I've already done some research and was pleasantly surprised.

As Blannie suggests, research the market first, then think in terms of a buyer and search for places to sell them. Given the nature of them, there might even be a market at art fairs, and they'll be in season fairly quickly.

You might have to find an auction house that sells these items and have them appraise the items.

If they're auctioned, ask what the auctioneer's % fee is; it might be so high that it's not worthwhile to hire a house but rather try to sell them yourself.

Another option for the lithographs, etc. is to inquire at a museum and see if it's interested in purchasing the items. In our area, the Henry Ford Museum at Greenfield Village has a treasure trove of merchandise, ranging from trains and classic cars to glassware.

And if you think in terms of a purchaser, where would you go to find these items? Then contact those places and see if they're interested in purchasing.

If your mom’s collection seems too small for a private sale they will combine it with another estate.

Auctions are a lot of fun. Go to one or two and see what you think. You might not find it so tedious if you enjoy going to them. Many auction houses have appraisers who will come out and give you an estimate.

With cell phones it’s very easy to snap photos and email them to different places.

If you are just wanting to make it go away you could call a thrift store to come pick it up.

Some consignment stores will take a look, pick out what they think will sell and you can donate the rest.

One of the very first things I'd do would be to Google the artist/book titles/paperweight types and see what their work is going for on eBay or auction sites. I have some artwork and was curious myself. Art auction sites will give you good info. You may have to set up a profile, but that's all. You can set alerts for the artists you want to follow. That will give you a better idea of what might or might not be valuable. I found some things I thought might have some value (Hummels and Wedgewood that my mom had collected for example) are of minimal value these days. It's a place to start.