How nature oriented? Do you like to camp? If you don't mind a couple of days in a tent, the best deal in Sydney is on Cockatoo Island, right in the Harbour, with a great view of the city. I think you can rent the camp spot for $45 a night, and if you need the tent and equipment, it's $75. The camp site overlooks the water. It's accessible by ferry. The ferry wharf has a little cafe.
April might be a little cool, but that's all relative - if you are used to winter camping, it won't bother you at all. We haven't camped there, but we know several Sydney families that have done it just for fun, and they had a great time. Cockatoo Island was once a prison, and a shipyard, among other things, so it's fairly industrial - not quite like camping in the bush.
Near Sydney, the Southern Highlands are also beautiful, and less "touristy" than the Blue Mountains - you won't have tour buses and rented vans and crowds all around you.
For Melbourne area, we had a great time on Phillips Island at the All Seasons Eco Resort - very comfortable little cabins, but it's a little like a mini-subdivision of resort cabins, so perhaps not to everyone's liking. But it's only a couple hours outside Melbourne, and we loved watching the penguins come ashore at night.