I'm from NJ/NY - do what L&B suggested. Travel on public transport to a location in NJ with a nearby car rental facility - even if it means one of you has to wait at the train/bus/ferry station while the driver picks up the car. You don't want to drive through NYC just to get to NJ for a day (by the way, what is the appeal of NJ - I'm curious as I grew up there and never thought people thought it was a worthwhile stop while visiting from overseas).
If you are looking for activities outside the city - it's not NJ, but the train service known as Metro North goes all over Westchester and you can do some lovely day trips just using the train/taxis at the other end - or LIRR out to Long Island.
OH - and get a SatNav - I think they are great - allows you to focus on which side of the road to stay on while "someone else" worries about the directions. I found that purchasing a basic model in the US is cheaper than renting one through car rental companies when renting for longer than 3 days (I think).
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