It's difficult to identify any individual part of Tallinn as a favorite attraction - the attraction itself really lies in the complete package - the Tallinn experience. I get the feeling that since being designated a UNESCO World Heritage Site, Tallinn has begun to attract a lot more tourists, but because of it's 'off the beaten track' location, things haven't yet snowballed into a tourist trap - go there soon before everyone else does.
Quick Tips:
Take your time - in the Old Town, no-one is in a rush - there's no need.
Best Way To Get Around:
In most cities you are unlikely to go and explore the suburbs and Tallinn is no exception.However, the Old Town has all you need to keep you occupied for two or three days and everything is within 10 - 15 minutes walking distance.
It's not really a traffic friendly city, particularly downtown as there are major long term roadworks throughout the centre (not in the Old Town) - appparently more tram lines being installed. However, if you do need to escape the city for a while, bus and rail links are excellent and Taxis are very cheap.