Picnic in Central Park
Nooyawka makes a v v good point. But first the food suggestion: on the east side, Dean & Delucca is at 85th and Mad; you can get just about anything you need there and it will be excellent. Proceed into the Park and you can find some great spots near the Great Lawn with shade (can be important in July). And, best of all, there is a loo just as you enter the park at 85th. BTW, dunno when you are coming, but check out the NY Philharmonic schedule in CP; you might be able to double dip at a most reasonable price (free).
Sables on 2nd Ave. bet. E77/78 for lobster salad, shrimp salad, and H&H a few blocks up for bagels. Then over to Turtle Pond for a view of the Observatory while basking in the sun. If you're here when Shakespeare in the Park is on, you can picnic in line while you wait for tickets. Andy's deli will deliver to the line if you don't feel like hiking in your own food.
Sable's Smoked Fish
1489 2nd Ave, New York, NY 10021
H&H Midtown Bagels East
1551 2nd Ave, New York, NY 10028
I really like all the suggestions you've gotten already. I think those are all really solid choices food-wise.
Rather than think of good food around Central Park, I've been trying to think of my favorite Central Park spots, and the best (of sometimes limited) food options in those areas.
The Pinetum is a beautiful spot, and I have fond memories of a great Zabar's/Pinetum picnic I had a few years ago.
Harlem Meer/Conservatory Garden (105th/5th Ave)--grilled chicken over lentil puree, phyllo cheese roll, hummous, baba ganouch, freshly baked pitas from Moustache; pretty good (not earth shattering) Mexican from El Paso or Lolita's.
The "Pool" at 101 on the west side--hmm, decent sandwiches at Cafe Amrita? Stuff from the upper west side Whole Foods or Trader Joe's?
There are also now more and more food options in the park itself:
At 72nd St on the west side, there's now a Pain Quotidien, a chain which has decent (if not particularly inspiring) sandwiches and other foodstuffs.
At the old Tavern on the Green site there are some food trucks set up. Recently had some great (but expensive) ice cream from Van Leeuwen http://www.centralparknyc.org/visit/things-to-see/south-end/tavern-on-the-green.html
I've also seen Wafels and Dinges carts set up: http://www.wafelsanddinges.com/location.html
El Paso Taqueria
1643 Lexington Ave, New York, NY 10029
301 W 110th St, New York, NY 10025
Le Pain Quotidien
50 W 72nd St, New York, NY 10023
1621 Lexington Ave, New York, NY 10029
57 Lenox Ave, New York, NY 10026
808 Columbus Ave, New York, NY 10025
2075 Broadway, New York, NY 10023
have your picnic on the hill to the west side of the fountain/pond at 72nd st. Less crowded and more picturesque than Sheeps Meadow
Food recs provided are great, add Citarella at 75th and Broadway and Fairway at 74th and Broadway or good deli counters and other prepared foods
2135 Broadway, New York, NY 10023
2127 Broadway, New York, NY 10023
2nd that and might add Grandaisy (room temperature roman style pizza), Bouchon Bakery (some bouchons or doughnuts or TKOs), Jacques Torres (maybe some chocolate chip cookies), Magnolia Bakery (icebox cake). And/or Shake Shack.
You could also go to Whole Foods if you really wanted.
I'm jealous already!
10 Columbus Cir, New York, NY 10019
176 W 72nd St, New York, NY 10023
285 Amsterdam Ave, New York, NY 10023
200 Columbus Ave, New York, NY 10023
366 Columbus Ave, New York, NY 10024
Well, the park is a huge place so where to send you for picnic goodies will vary a lot depending on where you want to be and where you are coming from. That said, one good place to go would be Zabar's, a NY institution all on its own. You could then enter the park at 81st and be near the Pinetum and the Great Lawn. (You should probably look at a map of the park so you can figure out at least a general area where you want to be.)
2245 Broadway, New York, NY 10024