look online. check MTA or metrolink
http://www.mta.net/ they have a trip planner on their website. i believe there is a metrolink station or subway that is in front of LAX which will then go directly to Union Station, which is what i assume you are referring to when you say Amtrack station.
$40 for a cab if youre lucky, that's all im saying. cabs in LA arent cheap. $40 will probably get you to the other side of the airport.