Vienna
Hofburg Palace area
Treasury (Schatskammer) - Make this your first stop, the Treasury where you can see the amazing crown jewels of the Habsburgs. Feast your eyes on bejeweled crown of the Holy Roman Empire, an embroidered red velvet coronation robe draped with ermine, golden orbs, a whopping big emerald, golden rose bush, reliquaries crusted with jewels, ceremonial swords and scepters, tiaras, coffers with bits of the true cross, and a royal crib. Free audio tours in English, and the Treasury is less crowded if you come earlier in the day.
Arms and Armor Museum (Neue Burg, Heldenplatz) - Check out the cool collection of arms and armor - shiny armor for tournaments, armor from the 1400s and 1500s for use on horseback, fancy "costume" armor (not for war or tournaments), two sets of boy's armor, armor for horses, plus crossbows, swords, helmets, and pointy metal shoes.
Kunsthistorisches Museum (Maria-Theresien-Platz) - This is a biggie, similar to the Louvre, chock full of Greek, Roman & Egyptian goodies, as well as masterpieces of European painting. In the Picture Gallery, kids will have fun with Winter and Summer, paintings where people's faces are constructed of fruits and vegetables by Arcimboldo, a Madonna with two adorable babies by Raphael, and the best collection of paintings by Pieter Bruegel, such as Hunters in the Snow, Children's Games, and The Tower of Babel.
Schmetterlinghaus (Imperial Butterfly House) - Once a part of the royal palace, wander through an indoor tropical garden, filled with exotic butterflies, colorful flowers, little waterfalls, ponds with fish. Open daily, year round (and a great place to nip into if the weather is cold and rainy).

Horse-drawn carriage ride - Pick up a horse and carriage (fiaker) in front of New Castle, and pretend you're one of the imperial family, riding around Vienna.
Spanish Riding School - Horse lovers in your family will want to see the glorious Lipizzan stallions in performances at the Winter Riding School. The Winter Riding School was built in the 18th century by one of the Habsburg emperors, and the gilded Baroque setting for this amazing show is unforgettable.
There are no performances during some summer months, and shows in June sell out, so buy tickets well in advance. Kids under 3 aren't allowed, but children ages 3 - 6 are free, if the child sits on the parent's lap. Another option is to buy tickets to watch the morning exercise (kids 6 and up).
Lipizzaner Museum - If you come to Vienna during summer, and the stallions are out of town, taking it easy down on the farm, your best bet is to head to the Lipizzaner Museum, across the street from the Winter Riding School, in the Stallburg (Stable Palace). The museum has film of the Lipizzaner show at the Winter Riding school, paintings of horses, harnesses and uniforms, and a gift shop where you can purchase charming stuffed animal Lipizzaner horses.
Volksgarten (People's Garden) - If you have little ones in tow, head over to the Volksgarten. In summer there are two sandboxes with water pumps for water and sand play and shade to lie out on the grass. This quiet park with rose gardens and fountains is a great spot to bring sandwiches and each out on the benches in warm weather.
Concerts - From May to October, listen to the sounds of Johann Strauss and Mozart, performed by a full orchestra in the Hofburg Palace, every Tues., Thurs. and Sat. A formal setting, these concerts aren't for little kids who get wiggly, but will be truly memorable for kids who are musically inclined.

If you don't make it to a concert, in summaer all around the Hofburg, listen for buskers, violinists playing classical music in the shaded hallways.