In August, I will be sailing from Port Liberty for the first time. I will be driving to and parking at the port. I would appreciate the procedures once I arrive there to park.

Specifically,

When is the best time to arrive?
When I do arrive is their a drop off point for the luggage at the terminal before parking the car?
Once I find a parking spot in the lot, are there shuttle vans or busses to take you back to the terminal?
On the reverse side, when we arrive back to the terminal, are their shuttles back to the lot?
Do we leave our family at the terminal with the luggage and go back and pick them and the luggage up?

Your experience in this regard will be appreciated

We're going out of the same port, but flying in and taking a taxi. Call your cruise line and they can answer these types of questions for this particular port.

Parking at Cape Liberty is adjacent to the cruise terminal, os there is no need for shuttle service, eyc. Here is a link to specifics about how to get to the port and parking details:
http://www.cruiseliberty.com/dirpark.htm

Thank you! I will just go with the flow the day of the sailing. It will be nice at the end of a cruise to get off the ship and just drive home. On a Sunday morning it should take less than an hour to get home.

They do have a shuttle to take you to the terminal and vice versa. You can drop your luggage off, then go back around and park.