^ Real NY pizza is found throughout the city but its hard to name a place - basically look for something on a side street or nestled among other affordable stores (hardware, bodegas, mom & pops, etc). Some of the most important elements of authentic NY pizza is that its simple, affordable and quick.
If you want shopping a fun area is 5th Ave from 59th Street down (starting with FAO Schwarz), and along 57th St in mid-town. Fifth will have high end gems like Bergdorfs, Bendels and of course Saks, and the expected chains like Sephora, Nike and Victoria's Secret. If you are walkers, you can take Fifth Ave all the way down to the Village. 57th St is more the designer chains: Channel, Louis Vutton, Tourneou, Tiffany, as well as Carnegie Hall. For more affordable shopping check out the Herald Square area. 34th St is home to lots of mall-type stores: H&M, Steve Madden, Levi's, plus there's Macy's and a full on mall.