Wife and I typically park at the village owned municpal parking adjacent to the Croton-Harmon Train Station, north of the city, just off of Route 9A. The lot is easy to acces, well patrolled by the village PD, relatively inexpensive, you can catch the Metro-North into Grand Central Station every 30 minutes or so, and minimal weekend cost. AMTRAK to Penn is also available here, but much more expensive, and doesn't run as often.
As for tours, I'd recommend the Grayline tours, the double decker busses where you can sit top-deck and see the sites are great for orienting yourself to the city. They also have a FANTASTIC around Manhattan Boat Tour which runs completely around the island. It's a bit more expensive than just a Harbor Tour, but worth it.
If you have minimal time, and just want to see the Statue of Liberty up close, but not stop, take the Staten Island Ferry from Battery Park to SI and back. Price is great - it's FREE! Take a few minutes in SI to wlk-up the waterfront to the SI 9/11 Memorial. Nicely done, and it offers a great view across the harbor to lower Manhattan.
If you want to goi to Ground Zero, you need to go on-line ASAP to get the fre tickets. No ticky - no entry. Still lots to see in that area, even if you don't get tickets.
As for other sites, places to eat, must see's, or any other questions, fel free to ask away! (I've spent a LOT of time in NYC fo worlk!)
2014 RAM 2500 Laramie 4x4 Quad Cab w/Cummins Turbo Diesel, 2009 Rockwood 2607, and 3 'yaks!
300W of Suntech Solar, a Rogue MPT-3024 Controller, and a Xantrex PW2000 Inverter