I'd recommend the new Crescent Park or taking the ferry across the river for the fireworks. For New Years, the quarter will be very crowded so do not drive and plan on walking a lot - I'd skip the rental. Most of what you'll want to see is accessible by public transportation. One nice tradition is to take in a fancy dinner at one of the old classic restaurants before the countdown, but be sure to make reservations early.
There is plenty of outdoor activities and parks, but depending on the weather, I'd add the WWII museum and surrounding art galleries and museums to the list.