Also google Southside Community Services as well (if you are not on the belconnen side of town), they have several centres around the area (one is for employees only DFAT-Currawong) Some lovely smaller centres as well as a larger one in the city. Do start ringing now and put your name down you can always take your name off when you visit if it doesnt feel right.
Also the other centres i can highly recommend are through Anglicare they have Calwell and Gordon Early childhood centres as well as Capital Hill (at parliment house for anyone who works in the buildig only)
Again depending on what side you live in a centre in Queanbeyan could also be an option, however i don't have any that i personally knwo about to recommend, but its another option
Good luck, Canberra is a gorgeous place and i hope you love it!