Arthut gives good advice and Veneiros,UPN,and Luzzos are all within a block of each other .I disagree however about Veneiros.I think it's one of those places that is coasting on an old reputation, for my tastes the best partry shop in New York is French and in Soho (walking distance from the Pizzerias,if you enjoy walking ):http://www.ceci-celapatisserie.com/flash/index01.html
I HIGHLY recommend the Poire William Gateau !!
My suggested downtown itinerary would be:
1) Stop by 'Veloce' on 2nd Ave for an aperitif. ( Tasteful modern design Enoteca )
2) Walk over to Luzzos on 1st and order a Margherita.
3) Walk or get a cab down to Soho (Broadway/Prince cross streets) and get Dessert @ Ceci - Cela.
4)Walk back along Prince and end up @ the old chestnut of a bar called Fanellis (Prince/Mercer cross streets)
A warm ,cozy long established NY Bar that more or less has avoided becoming a tourist trap despite being at the epicenter of this trendy neighborhood.
Have a good trip.