The J train is a great way to get downtown from JFK too - no need for a cab.
The subway in general is good to get to and from JFK. I've used it on all my trips and found it to be the best value option, costing a standard fare + $5 for the air train (compare that to LHR!). But I've never had too much luggage. Going into the city is never a problem, there's always been plenty of room on the A and E trains (never tried the J/Z option). Coming back can be more difficult because you'll probably be boarding in Midtown, the busiest part of the network - you will not be popular if you're taking up multiple seats with big suitcases during the rush hour!
I've been to NYC twice and both times used the subway as almost my sole means of transportation, including airport transfers. I had no problems at all during either of my visits. It is a fantastic system.
, the line will provide quite a lot of relief. Many of those from the Bronx and the Upper East Side who currently stay on the 4,5 & 6 lines and want the southern end of Broadway and Lower Manhattan will change at 125th St and use the (more modern) 2nd Ave. 'T' line. Those who want Broadway, Times Square, Herald Square etc. will change to the Q line. This will lighten the load on the Lex lines below 96th, giving more space for both Metro North and East Side Access pax who alight at Grand Central. That's not to say that something akin to a Crossrail line wouldn't be a valuable contribution to travel into and out of the city.