Well, between mid-June and mid-October 2012, Rowan and I spent 3 weeks in Hong Kong, Taiwan, Japan (Hokkaido), and 3 months in Europe ... all without a car.
http://www.machka.net/RTW_2012/RTW_2012.htm
We did acquire vehicles for the remainder of that trip (Canada, US, Western Australia).
We haven't done too much travel since we returned to Victoria in early February 2013, but we continued to cycle here, there, and everywhere through the summer and autumn (evenings, weekend, hub-and-spoke tours), and are looking forward to getting back into that sort of thing come spring.
Now last weekend, we did a little touristy adventure. We drove our van to the closest city train station, parked it there, and took the train into Melbourne. Then we spent the afternoon touring Melbourne by tour bus and tram. The cycling bit was stopping off at our favourite cycling shop.
If you were "car-free", Melbourne would be fairly easy to get around by train and/or bicycle. And like many bigger cities, it has guided tours (tour buses (hop-on, hop-off), trams, boats on the river, walking tours etc.). Or perhaps you might prefer to cycle the 100+ km of cycling trails (which we've also done).