For casual visitors, the Newberry offer a range of temporary exhibitions (free admission), as well as a small selection of items on show from their permanent collection. There's also a good bookstore as part of the souvenir store.
I was primarily at the Newberry to undertake some academic research and found the staff consistently helpful and engaged - but you do need to arrange with them in advance if you want to use the library facilities and register (there's no charge). Rules are strictly enforced, so you can't take bags or food in (though there are free lockers) and there's a small coffee station on the ground floor, so you don't have to leave the building for refreshment. My only disappointment is that the catalog is a little patchy, and it's not always easy to determine what to get from the stack.