Pretty much, Vienna is awesome with kids. You can take a stroller anywhere (ANYWHERE) and if there are stairs or other obstacles, complete strangers will help you carry the stroller where you’re trying to go.
Children are welcome pretty much everywhere — at museums (although tickets can get expensive), historical sites, restaurants, public transport, festivals, etc. I haven’t taken my kids to a single place where I felt like we were out of place. Generally, the Austrians are pretty indulgent of kids (although the like their dogs even better).
I’ll write more later … there’s so much to say. But, Vienna is great for kids.