Actually the Verazano Narrows Bridge will put you in Brooklyn, which is technically the end of Long Island, BUT, you are left with nothing but congested city streets to get you further out on the island. You cannot use the Belt Parkway or any other parkway in NY for that matter due to low bridges. I'm not familiar with what roads you would need to take from there.
Anyone I have heard about uses the Geo. Washington Bridge, but as others have said, the is NO way to get there without a ton of stress. Unless maybe one of the ferries from Connecticut. I have no experience with those either.