metro.net is indeed your information source. You'll ride the Red Line (subway) from Union Station to the 7th Street/Metrocenter station across downtown, then the Blue Line light rail from there to Long Beach. Blue Line in particular can be very crowded so having much luggage could be a problem - sending the big stuff ahead via FedEx is an option.
Also, be aware that rail transit in LA, as in most of the US, doesn't run all night, so if you're delayed and arrive at 3AM you may be stuck unless there's an all-night bus ... again, check metro.net. Or Uber, but a ride to Long Beach might just bankrupt you. And LA has gone entirely to electronic fare payment on rail lines, so you'll have to buy a card (fare + a card fee) from a machine at the station and "tap" it on entry and exit at a reader. Transfers, in general, are just another fare.