Can I take snacks on an international flight?

I am flying from Melbourne, Australia to New York via Sydney, San Francisco and Los Angeles. Can I take snacks on board? What about food from home like a sandwich? I will have to get rid of them at San Francisco when I go through customs, right? Or will they not even make it through Sydney? The last time I went to the US there was border protection or security before we got on the flight where everyone's carry on got searched, patted down etc, would food get through that? I was flying with an airline last time that gave at least 2 meals that time so I didn't have to worry. We only get one meal on the 14hr flight to San Francisco and it's during the day so I'm going to want something to eat that I don't have to buy for some insane price. Thanks heaps if you know anything.