What is the first record of a "museum" for the purpose of displaying old objects, and what objects were in that museum?
The oldest museum known to exist is Ennigaldi Nanna's museum, which was in about 530BC in Ur (Iraq). It had old Mesopotamian objects in it.
The idea of a museum really began to flourish in the Renaissance. The famous Uffizi was first opened to the public in 1582. his held mainly artworks. At the same time, may royal and aristocratic families were developing private collections, mainly consisting of artefacts from the ancient world (e.g. the Roman Empire).
This developed into museums as we know them today, with early offerings including The British Museum (opened in 1753) and the Louvre (opened 1792)