I use the Flyaway To/From Union station regularly.

Here are a few tips:

ARRIVAL AT LAX
When you ready to exit the terminal, go to the lower level/Baggage Claim and exit the doors. You will be at curbside, where only taxi and private vehicles stop. You need to look across that line of traffic to the islands where ground transportation stops. Look up and walk until you find the GREEN sign that states FLYAWAY, BUSES, LONG DISTANCE VANS (One at every terminal), and then cross the street to that island and wait for the coach. There are three different Flyaway buses, and many times they are NOT clearly marked/signed. The coaches sometimes say Americoach who is the contractor that runs it. Ask the driver to confirm UNION STATION when they stop, and sometimes you may even need to flag the bus down. They take your luggage and you board the bus. The driver will not collect the fare. When you arrive at Union Station, you will be let off and asked to buy your ticket at the kiosk for $6 each One Way. You return to the bus/driver, hand him your ticket, and you are given your luggage.

DEPARTURE TO LAX
Arrive at least 5 minutes before scheduled departure. You will this time pre-pay your ticket at the same kiosk, hand the ticket to the driver to check your luggage and board. You are dropped at the Upper Level/Departures upon arrival, curbside.

Web site is below:

LAX Flyaway