How to travel by public transport form the city to Killarney Heights?

Forest Coach Lines run the private bus service from the Forest to the CBD, Chatswood, Warringah Mall and the Northern Beaches - you still use your Opal card but the buses are generally in better condition and cleaner than other city buses. To get the schedule just look at http://www.forestcoachlines.com.au/timetables/270-271-274.pdf - note that there are express buses (marked with a "K") that originate in Killarney Heights and go straight to the CBD - so you should easily get a seat and then it's 30 minutes from Jamieson Square to Wynyard in the morning. That's pretty quick, but it depends on traffic, of course. Sometimes it's actually faster, in my experience!