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Disneyland Resort Express bus doesn't run to Long Beach. Only to LAX and SNA. So your options are a shuttle or private car or van service, or a taxi. There are supposed to be "taxi vans" but I'm not sure of their availability at LGB. I would call the airport and ask.
Super shuttle from LGB costs $35 for the first person, then $9 for each additional person over the age of 3. So if your kids are all over 3, that would cost $71. However, there's a coupon here:
http://www.anaheimoc.org/Articles/Archive/Webpage10153.asp
for $2.00 off per person, each way, when you book with Super Shuttle online. (it's the very last coupon at the bottom). That would bring it down to $61, plus tip.
Edit: I just saw that your youngest is 20 months, so that would be just a charge for 4 people, so with the coupon discount it would be $54 plus tip.
If you can get a taxi van for the $40 flat rate, that would be a better deal.