New York/The Hamptons?

I don't recall a discussion beyond NYC hotels but can't search at the moment due to the function being disabled for now.
I'm planning a NY trip and for the first few days around July 4th weekend being really touristy for the Daught's benefit - staying on HH points at the Hilton Times Square. After that, I fancy going off to the 'Hamptons' in a rental for the next three weeks, but I'm overwhelmed by the property choices and wouldn't know if an area was good, bad or indifferent.
Or should I book an apartment in Manhattan? Or somewhere else? Budget would be IRO $3000 per week, more for somewhere amazing. Private pool would be nice, like being able to walk to restaurants/beach but will probably have a hire car if out of NYC.
Thanks in advance for any ideas.
Regards
RedV
I'm planning a NY trip and for the first few days around July 4th weekend being really touristy for the Daught's benefit - staying on HH points at the Hilton Times Square. After that, I fancy going off to the 'Hamptons' in a rental for the next three weeks, but I'm overwhelmed by the property choices and wouldn't know if an area was good, bad or indifferent.
Or should I book an apartment in Manhattan? Or somewhere else? Budget would be IRO $3000 per week, more for somewhere amazing. Private pool would be nice, like being able to walk to restaurants/beach but will probably have a hire car if out of NYC.
Thanks in advance for any ideas.
Regards
RedV