Completing the True Guertena Exhibit
Checking paintings gets them into the true Guertena exhibit (you have to be able to read the full name, though, so if the name contains words that Ib doesn't know, you have to have Garry with you to read them to Ib or it won't count). Some paintings, however, require that you read books (by examining bookshelves). Ones that contain unknown words will require Garry's assistance, obviously. To get Garry, Mary, and Ib's mother and father into the exhibit as well, get the four good endings (where Ib escapes through the Fabricated World painting). It doesn't matter if you get an ending before unlocking the true Guertena exhibit; as long as it's on the same file (continued through new game plus(es)), it will count.
Garry - Get the Promise of Reunion ending.
Ib's father - Get the Forgotten Portrait ending.
Ib's mother - Get the Memory's Crannies ending.
Mary - Get the Together, Forever ending.
New Game+
Beat the game once (getting an ending where Ib escapes) to unlock new game+. It will have a bonus dungeon that you can reach by going down the new stairs south of the reception desk. Completing the bonus dungeon gets you the Black Key, which lets you into the True Guertena exhibit upon getting another ending where Ib escapes.