If you have something that must be checked, such as a bicycle, or that you really want out of the way for the trip, you can take a Metrolink train to the Los Angeles station from the same Burbank airport station that Amtrak uses, on weekday mornings at least. The Amtrak Starlight and the Surfiner trains use the south train station at Burbank airport.
Weekends, it's more complicated – Metrolink trains don't run through the south Burbank airport station. There's one train that runs through the new north Burbank airport station and the downtown Burbank station to LA.
You can also take a Surfiner to LA, but you'll run into some of the same luggage restrictions.