bennos

With that many people and bags, you might consider taking a car of some form into the city, and then using public transit after you get there. You could prebook via a service like Dial7, though probably Uber/Lyft would work too.

Otherwise, the Salisbury Hotel is right near the 57th St (F) Station on 6 Av and the 57th St (R) Station on 7 Av, so transit is pretty easy: take the Q70 Select Bus Service to Jackson Heights (just get off where everybody else does), then go downstairs and take an (F) or (R) train inbound to Manhattan. The (F) is an express and the (R) is a local, but the actual time difference is maybe 5 minutes. The (R) will most likely be less crowded.

I believe there are Metrocard Vending Machines in LGA in baggage claim, but I don't remember exactly where. You would buy your 7 day card from there (or maybe from a newsstand). If you can't buy these in advance, you will need to buy a cash fare for the Q70, and only coins (including dollar coins) are accepted. Then buy your 7 day card when you get to Jackson Heights. (There is no bus-to-subway transfer for cash fares.)

The Q70 is a "Select Bus Service". You need to pay your fare before you board, from different vending machines. Your 7 day card is not directly valid for transport, you have to use your card to obtain a proof of payment receipt. See more info here.

At the subway, my suggestion would be to send one or two people through the turnstiles, then pass the luggage through, then have everyone else go through. It's three flights of stairs down to the platform, though there are (really slow) elevators.