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It will be less expensive to rent a car for a day than to pay a shuttle!
It may cost several hundred dollars but not completely sure.
Why are you staying in Long Beach??
There may also be train stations closer than LA.
You could take the free Passport shuttle from the Queen Mary to the transporation center in downtown Long Beach to its end in downtown Los Angeles. Then take the DASH from where the Blue Line ends to the Train Station. Private transfer through Supershuttle would be $68 (plus tip) for all four of you.
If you are comfortable driving a car in the US, I think it would be the easiest and cheapest; plus, it would give you the opportunity to stop along the way at Laguna Beach, Mission San Juan Capistrano; and/or La Jolla. There are several rental car companies in downtown Long Beach. Google car rental 90801
The on-line quote from SuperShuttle is $391.96 for a Towncar and $445.45 for an SUV.
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