Wave Hill, a city garden that frequently has music. Great for picnic lunches and resting your tired "dogs".
https://www.wavehill.org/
Shakespeare in the Park (free) will run through sometime in August. Grab a picnic meal from Zabar's (a Westside institution) and nosh as you wait in line.
https://www.publictheater.org/Free-Shakespeare-in-the-Park/
http://www.zabars.com/on/demandware.store/Sites-Zabars-Site/default/Link-Page?cid=OUR_STORE_ON_BROADWAY
A rarely seen by visitors or even residents, formal garden at 5th Ave and 102nd Street. Beautiful statues, water features and wisteria trellis. Not what you'd expect to see in Manhattan.
Of course the Cloisters. I take my DGD there least once a month and she's still not tired of it- in fact there's a pic of her sitting in front of part of the Unicorn Tapestry absorbing the details in one of the museum's magazine. Schedule a trip there same day as the Metropolitan and you only pay an entrance fee for one of the two.
Brookfield Place which is just across the street from WTC yet remains a destination in itself. Fantastic water views, great eats, people watching and year round activities. During the summer they like to show movies at night outdoors. Grab a prime rib panni from Eataly's and go to heaven on earth. Or take an outdoor seat at Le District ('spensive French bistro but great night scene) or PJ Clarke's. PJ's raw bar is fresh, made to order and the seafood platter is quite filling although my piggy eyes thought it wouldn't be. Wrong (thank goodness for my waist line, LOL) and a fine light supper or lunch.
Blue Smoke across the street is a BBQ haven and the brisket tips, collard greens and mac + cheese are truly worthy. I swearrrrrrr I only ate a tablespoon of the mac......
High Line Park, on the edge of the Westside, is one of our newest "got to be there" places. It's on the site of a previous railroad and rises above the street in many areas. Great landscaping, eating and people watching. Also you can see the aerialists' school attendees working out and gosh many are quite good.
Bring your camera and walk over the Brooklyn Bridge around sunrise and eat breakfast at Junior's once you get there. Take the subway back to "the city" and avoid the hordes of tourists and city folk crowds. Make a stop at Federal Hall and checkout the newly discovered parts of the wall (embedded in the street) that gave Wall Street it's name and delineated the upper reaches of 17th century New Amsterdam. Feel like exploring more of Brooklyn? Walk (or grab the subway) up Flatbush Avenue make a left on Eastern Parkway to the Brooklyn Museum. Sorry you'll miss the Georgia O'Keeffe show:(.
https://www.brooklynmuseum.org/
Since you are staying in Staten Island take a trip over to Sailors' Snug Harbour. Haven't been there for a while but the memory lingers:).
http://www.noblemaritime.org/sailors-snug-harbor/
Take in a game at Yankee Stadium and then stop at Giovanni's for brick oven pizza. Should you still have some energy walk a few blocks up the Grand Concourse, to the Andrew Freedman House. It was built by a millionare for formerly wealthy people who lost their fortunes. Each resident had the use of 3 servants so they could better remember the "old days".
http://www.ohny.org/site-programs/weekend/sites/andrew-freedman-home
Take a walk through Greenwich Village. It's nothing like I remember it anymore (gentrification....sigh ish ish) but the historical spots, still remain as well as the twisting streets. You can still close your eyes and imagine Edith Wharton, O. Henry and Bob Dylan walking down the streets.
http://www.tabletmag.com/jewish-news-and-politics/76248/buried
HTH and have fun!