I particularly love the idea that paying $15 in subway fares (and, really, even less given that the OP made it clear they'd be buying 7-day cards) is somehow a massive financial burden relative to the premium paid to get a car service large enough to seat 6. That's going to be at least $40 and probably more.
If you're comfortable as a family using mass transit then this particular trip is a piece of cake. The hardest part will be negotiating the turnstiles at the subway station with the bags. You've got enough people to have one go through and then just hand the bags over/under until everyone is in.
It is very clear on the bus the stop to get off at for the subway transfer and then just head downstairs to the Manhattan-bound platform. Enjoy the visit!