City Bike with Ortlieb
Cycling is getting fashionable. Cities like New York are emerging recently as bike friendly towns. That was not always the case. AbouttTen years ago, you were not prepared to see any bike at all in Manhattan, and it was too dangerous to ride one. Today, there are many seperate bike lanes in Manhattan, but you still need to wear a helmet and be cautious with the cars and especially with the cabs. There are bicycle shops sprouting everywhere in New York, and you can get rentals.
There is now also a bike map for New York, and also Google maps are testing bike maps. There are foldable bicycles which are very practical to carry in your car. You park your car where you find a parking spot, and then you are off on your bike to go shopping in the city. 
Buy some bicycle bags, the ones from Ortlieb are of high quality, and they are all water proof. You need water proof bags. Bags which are only water resistant will not do it, because if it rains the content will be wet after about twenty minutes. With Ortlieb bags the content does not get wet at all. I also use Ortlieb bags to go to the beach.