If you fly with Skyteam, your routing would probably be

LAX-[ATL]-EZE-[AMS]-CPT-[JNB]-[NBO]-BOM-SGN-SIN-[SGN (or CAN or ICN)]-SYD-[NOU]-PPT-LAX.

Places in []-brackets would be places where you change planes, places in ()-brackets are alternatives to the airport mentioned right before. The trip from CPT to JNB is, as far as I know, not possible with Skyteam, so you would have to do this trip with a different airline, maybe some low-cost carrier?

In general: I recommend that you go to the websites of the three airline alliances and look up your trip on their timetables. Alternatively, go to a travel agency.