Ringstrasse, is sufficiently small to be walkeable, but if you want to go further, in particular to the Schonbrunn Palace, you're going to need transport.
You could take taxis, and a couple of times we did and they were efficient and friendly, but the public transport is pretty comprehensive and cheap. We got a 72 hour pass that included all trams, trains and underground journeys plus a return ticket on the airport to city train for just 33 Euros.
The underground network might have been smaller than London's but it connected all the main locations. We did take a couple of trips on the trams (above) just for the experience - in particular to do a loop around the Ringstrasse.
There were horse drawn carriages for those that are into those sort of things (below) but we weren't so gave that a miss.