What to do in North Sydney area with a 1 & 2 year old?

[nala05]

We are moving to North Sydney for a month and I will be cooped up in a small apartment with a 1 & 2 year old.

Any suggestions of things for us to do?
Good playgrounds?
Play areas in shopping centres?
Things like Gymbaroo that don't require term memberships?

Any suggestions would be greatly appreciated!