I do this trip on average twice a month and for someone with children I would definitely take the bus - when you come out of baggage claim and clear customs there will be a large area that is well marked for the friendly limousine bus company. Just get in line and tell them (they understand English well) which hotel you are staying at and they will tell you the time the next bus leaves for that destination.

As for jet lag there are many strategies - and mine are from years of personal trial and error. For me, I sleep immediately when the flight takes off (I take a pill if not sleepy already) - I don't eat anything, just sleep. For me getting the body on the new time is critical - and that includes the stomach. I try and only sleep about six hours and then get up - when everyone else is sleeping lol - and then have some light food. I realize this will be challenging with kids - but good luck. Spending two weeks a month in Asia this is what works for me - the first day can be brutal but it gets better fasat after that. On the way back I try to not sleep at all (which is hard) - so you get back to the states exausted and ready to sleep - which coincides with the local time quite well as by the time I get home in NY it is about 8pm usually...