You are mixing up the palaces and the art museums. The Hofburg and the Schonbrunn are the two palaces in Vienna. The Hofburg was the "city" palace and the Schonbrunn is the "country" palace. Each of these palaces has numerous collections (Hofburg: Silver Collection, Imperial Apartments, Sisi Museum, etc; Schonbrunn: Apartments, Carriage Museum, etc). I would highly recommend that you chose to tour the collections of either the Hofburg or Schonbrunn as they are repetitive, but not both. Personally, I would look at the Hofburg from the outside and take the interior tour of the Schonbrunn. In addition to the interior, Schonbrunn has amazing gardens and the palace is lovely (the Hofburg's exterior doesn't even compare).
Now for art museums, Vienna has many - Kunsthistoriches, Belvedere, Albertina, Leopold, MUMOK, etc. Each has a distinct personality and collections of different genres. The Kunsthistoriches (KHM) is in a purpose-built, but beautiful musuem. This is one of the world's great art musuems and the collection ranges from the antiquities through the Middle Ages to the Renaissance and beyond. It is well known for its Bruegel collection. The Belvedere is known for its collection of Klimt paintings (particularly the Kiss) and is a beautiful palace in its own right. The Albertina has a mixed collection which includes Impressionists. The Leopold is a fascinating museum with a wonderful collection of Schiele and other Austrian modern art. The MUMOK is the modern and comtemporary art museum in Vienna. The Leopold and MUMOK are located in the MuseumsQuartier, a large complex of several museums, large and small. It is worth visiting the MuseumsQuartier even if not entering the museums as there are several nice restaurants and other attractions housed inside this formal royal stable.
You do note that you are not really into museums anyway. Also worth pointing out that you will be in Vienna at a lovely time of year, so maximize your outside time! Personally, I would visit Schonbrunn, the Belvedere and have lunch at the MuseumsQuartier. Here is a list of restaurants inside (my personal favorites are the Glacis Beisl and Corbaci):