The Szabó Ervin library is definitely in my top 5 places in Budapest. It is the main library, with thousands and thousands of books in Hungarian and other languages.
The house consists of two parts: a newer one, and an older one that was built in 1889, in a neo baroque style. It is basically a palace, and all the wallpapers, fireplaces, furniture and inlaid wooden coverings look the same as they did a hundred years ago.
Anyone can get a free pass into the library at the information desk, and can go inside to look around, read some books or magazines.
They also have a nice cafeteria right behind the entrance, where anyone even without a pass can go. It is in the old building with all the nice details. Everything is very cheap here: you can get a tea for about 150 HUF, or a latte for 300 HUF, and drink it in a place with a wonderful atmosphere.