We went to Vienna last year as part of our Austro-Hungarian adventure.
We were travelling by train and went to Vienna from Lake Balaton, and used the night train back to Amsterdam.
Vienna is a very walkable city and it has excellent, easy to use, public transport.
The palaces are huge, as are their grounds - you really need to devote an entire day to any one of them - we didn't and it was a mistake.
Agreed on the tram route around the Ring - it's easy to do, and there are a number of online suggestions for making it slightly more interesting.
We were going to visit the Freud museum, but after talking to a couple of people who'd been, decided to give it a miss (too dry without enough interpretation).
Unlike other similar offerings, we found the Vienna Card to be good value.
We found the night train exhausting - despite having a double, en-suite compartment. I didn't have a relaxing journey, finding the berth uncomfortable and the movement distracting. If I were to do it again, I'd programme in a day afterwards to recover (ie for *me* it doesn't save a hotel night) - and I love trains & train travel.