Make the cards into a book. I actually have a book of old greeting cards. I used an old photo album – the ones with the clingy plastic overlay – and inserted a card on each page. You can’t see the inside of the cards in this way so I was thinking how else can the cards be made into a book where you can see both the inside and outside of the card? Make the cards into a saddle-stitch book. I will eventually make one out of some cards I have at home in a bag and post later. But there are lots of great tutorials out there on how to make a saddle-stitch book. Here is one that I like: Saddle Stitch Binding. If the cards will end up being a really thick book then use the Coptic Stitch method: Coptic Stitch Binding or make two books (as the coptic stitch is very work intensive.
Frame as artwork. I wouldn’t do every single card you have this way but select no more than 5 of your favorite cards and frame them as if they were art. If you want the inside of the card on display as well; photo copy it, arrange and glue the card and copy next to each other on a piece of poster board or cardboard, cut to fit in the frame, and hang on your wall. In this photo, the person matted three cards in a single frame. The mat holes were cut the same size and the cards were floated in that space.
Decoupage. If you aren’t that attached to your cards but keep them for the image on the front, cut off the front of each card and decoupage them onto an item like a small table, canvas, wooden box, etc. There are many tutorials online for decoupaging. Here is a good very basic one for doing it on canvas How to Decoupage. (You could also still scan in the message part of the cards and keep as digital files.) In this photo, the person decoupaged cards on this end table.
The same idea can be done with old photos as I've done here to a kitchen table with old black and white photos from college. Make sure to apply a coat of polyurethane to protect from moisture at the end of your project.