Picnic tables are located by The Great Hill (West Side from 103rd to 107th Streets), the Ross Pinetum Playground (West Side at 85th Street), and at the entrance to the Park on 90th Street and Fifth Avenue.
Picnics of more than 20 people, require a permit through the NYC Parks Department. You can obtain a permit here.
Tables and chairs are not permitted to be set up on any grassy areas, due to the damage that the legs may cause.
Alcohol is not permitted in Central Park, except at specific restaurants and their patios in Central Park.
Barbecuing, open flames and grilling is not permitted in Central Park, with the exception of 3 days per year. For more information see: http://www.centralpark.com/guide/activities/grilling-or-barbecuing.html