Sydney Airport is easy with kids! When departing internationally there are separate security lines for those traveling with children that were much shorter than the regular ones. The family bathrooms have everything you could need: there were 3 separate rooms, one with a plush chair to nurse or pump, outlets and a microwave, one with a kids sized toilet, and one with a changing room. There is also a playground in the McDonalds in case you have some extra time and your kids need to burn some energy.
Getting From Melbourne Airport to the City
There is an express bus that gets to the city in 20mins, the SkyBus, which cost me $28.96/RT. It picks up in front of Terminals 1, 3, & 4. Even if you are coming into Terminal 2 the walk between terminals is extremely short as it's a small airport. If you are staying in a hotel in the city, they even have complimentary shuttles that will take you from the bus station to and from your hotel. Children under 4 ride free :-).