I was on the June 12 sailing of Empress of the Seas (8 day to Bermuda). I drove to Bayonne from Long Island, arriving at the pier around 2:00 PM. Baggage drop off and parking were very easy. Check-in was a breeze as there were no lines at all. You do need to take a shuttle bus to the ship as it is docked about a half mile from the terminal. All in all, I was aboard ship before 2:30.
Debarkation of course always seems to take forever, but this is the case anywhere. The ship docked around 8:30 AM (you need to vacate your cabins by 8:00) and cleared customs by 10:30. By that time they had just about finished unloading the baggage and trucking it to the terminal. The first passengers were able to debark around 10:45. The last passengers got off around 12:15 (I know, because I was one of them). They have 5 shuttle buses to take the passengers to the terminal and they generally kept up with the flow, although I think having one or two more would be optimal.
Having parked at the pier, I cannot comment on the availability of tranportation, but from what I have heard, it would be best to use either RCCL's or pre-booked private transportation. I simply walked to my car with my luggage (I was travelling alone) and drove home (little traffic until I hit the BQE, oh well, that's New York).
The cruise, by the way, was wonderful. it was especially nice to dock at St. George for a day, something that RCCL ships haven't done in years. Also, almost all the public areas on the ship had been refurbished and looked very nice.