I would get tired of just sailing in NY Harbor too.
My boat is up the Hudson just above the Tappan Zee Bridge at Nyack and it is pretty and fairly quiet up there. People anchor overnight off Croton Point or across the river at Bowline Point. This area is a good 7 hours travel by boat from 79th st. The water is brackish and for swimming I prefer the salty waters of the LI Sound.
You should be able to go around Manhattan and up the East River but you will need to check the tides and currents in something like the Eldridge Tide & Pilot manual so you don't battle an opposing current. It is actually a fun trip and a little bit challenging but you should be able to reach the City Island/Manhasset Bay area in around 4 hours. About 2 hours farther is Hempstead Harbor which has places to anchor or pick up a transient mooring for around $40/night.
Then there is Raritan Bay to the south. RYC probably has transient moorings as well. Maybe not so bucolic but not so far either.
Transient moorings at Nyack Boat Club are still $25/night and it is a pretty river town.
Good luck and have fun.
"The cure for anything is salt water~ sweat, tears, or the sea." ~Isak Denesen
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