I would suggest the Carnival Shuttle service. Although, It all depends on how much time you have and what your plans are. Are you flying in just hours before embarkation? The day before? Hotel plans? They all make a difference. LAX to Long Beach is quite the trek depending on what time of the day you're going too. The 405 freeway can be a nightmare so most taxi's will take that route just to bump up the bill. The 105, during the day is pretty quick, then down the 605 is your best bet.
p.s. I lived down there for over 20 years.
For a couple of years, I was a diesel mechanic. I had to travel those freeways to different locations to work on the trucks. Ya learn which freeways to take to save your sanity.