Take the shuttle back to the airport, then take the airtrain to Jamaica Station, you will have to pay a small fee as you exit. Then take the "E" train subway to 42nd, Time Square Manhattan. There are people at the subway station to help you with your fares. You can take a taxi or Uber from the hotel to Jamaica Station but I'm not sure of the cost.
I dont know about LaGuardia. We always fly out of JFK and use the park and fly package at the Hilton Garden Inn in Jamaica, Queens. So convenient. Great price. Their staff is so helpful and friendly, just call the front desk..they'd be more than happy to assist you.
I am not sure, because we did not rent a car to get around. When we were there there were construction in the parking area. Call ahead prior to booking a reservation if you will be renting a vehicle to get around the city. The parking there is extremely expensive.