Public libraries can be a great place for shows. Some libraries that have dedicated gallery space will have their policies about their gallery on the website. It's always a good idea to do a little homework before approaching a library of gallery, and if you can find a submission policy, use those as a guide.
I pulled together a show for my local library, and it has led to other shows in other venues... especially since it's a completed body of work, and framed, ready for presentation.
Those costs, by the way, can be considerable, but once invested, subsequent shows won't cost as much.
Good luck.