JFK International Airport AirTrain

Here is how you can get there:
1. By public transportation:
Take Airtrain from any terminal on track 1 to Jamaica Station( be careful - on that track there are two alternating trains going to different routes; check the sign above the doors as the train arrives; the red colored sign is Jamaica one). Then take subway line E to Roosevelt Ave stop and transfer to line F (going towards Manhattan). Get off at 14th street stop(cost-$7.25).
2. By Taxi:
Yellow cab (from JFK to any point to Manhattan) costs flat rate of $45 + bridge toll($5.50). The price is for the whole taxi, not per passenger.
3. Shuttle:
There is shuttle(look for it at Ground transportation counter or Welcome Center) that goes to any point in Manhattan for $18.50-$25 per person.
JFK_E

There is door-to-door shuttle called Super Shuttle Long Island that will take you to MacArthur Airport. You can book the shuttle at Ground Transportation(Welcome Center) counter. It costs $63 per person.
You can also get there by public transportation:
Take Airtrain from any terminal on track 1 to Jamaica Station( be careful - on that track there are two alternating trains going to different routes; check the sign above the doors as the train arrives; the red colored sign is Jamaica one). Then take LIRR train to Ronkonkoma. From the train station there is shuttle every half hour(cost-$5) or S-57 bus($1.50 fare).
Total cost: $5 + $8.25($11.50- peak) + $1.50($5 for shuttle).
JFK_E

You have plenty of time to go to the city and come back.
The public transportation usually takes between 1hr-1:30hrs(it depends on which part of the city you are coming from/going to).
Unfortunately most of the trains are under ground(only subway A has a part of the route that is above the ground).
If you take the express bus(NYAS) on your way back(from either Times Square or Grand Central) it will cost you $15.
There are no sightseeing tours that leave from JFK. You can book such a tour in the city.
Check out those links:
http://www.newyorksightseeing.com/
http://www.citysightsny.com/
JFK_E

Take Airtrain from any terminal on track 1 to Jamaica Station( be careful - on that track there are two alternating trains going two different routes; check the sign above the doors as the train arrives; the red colored sign is Jamaica one). Then either:
-take subway line E to Penn Station(total fare - $7.25;~1 hour) or
-take LIRR train to Penn Station(faster;total fare ~ $12, less on the weekends; 35-45 min.)

From Penn Station take NJ PATH train(Journal Square line) and get off at Pavonia-Newport stop($1.75,10min travel time).
Once you are off the train walk south on Washington Blvd - you will see Westin Hotel on your right.
JFK_E

Hello Susanna,
Once at Grand Central you can walk the distance - it is three-four blocks away going east on E42nd street.
Take Airtrain from any terminal on track 1 to Jamaica Station( be careful - on that track there are two alternating trains going two different routes; check the sign above the doors as the train arrives; the red colored sign is Jamaica one). Then take subway line E to Lexington Ave stop and transfer to subway line 6 going downtown. Get off next stop-Grand Central.(total fare - $7.25)
JFK_E

I won't suggest you to go on the public transport when you travel with bags and a baby.
A cab will cost you flat $50.50(including toll; for the whole taxi) - the problem is that you must have a baby seat(they do not provide it).
Door-to-door shuttle as another option , but it will be too costly - $18.50/person. If you still consider that, you can book the shuttle at Ground Transportation(Welcome Center) counter.
You can try a private service/car - they will charge you as much as a taxi and over the phone you can ask/arrange a baby seat as well. Just go to Ground Transportation (Welcome Center) counter and dial 31 or 34 on the self service phones.
JFK_E

You can either take LIRR train to Jamaica station and then Airtrain to the terminals(35-50 min ride) or take subway E going towards Jamaica, Queens, get off at Sutphin Blvd stop and take Airtrain to the terminals(50min to 1.15hrs ride).
JFK_E

Yes,it is possible, even the Airtrain will not take you directly to Penn Station. Here is how:
Take Airtrain from any terminal on track 1 to Jamaica Station( be careful - on that track there are two alternating trains going two different routes; check the sign above the doors as the train arrives; the red colored sign is Jamaica one). Then either:
-take subway line E to Penn Station(total fare - $7.25;~1 hour) or
-take LIRR train to Penn Station(faster;total fare ~ $12, less on the weekends; 35-45 min.)
JFK_E