I think there are some great things to do in October. It’s a great time to visit.
The leaves should be going golden in Central Park, so do that. The Metroplitan Museum overlooks CP, so I would go there too. It has wonderful American exhibits. One day each week it’s donations instead of the entry charge, so check that out.
Going to Brooklyn and walking back over the bridge is iconic. It’s really worth it. The other great walk is along the High Line. We went N to S, but it’s great for views and you can stop off at Chelsea market. It makes you realise what can be done with an old railway line! Fantastic.
We also like the Empire State and The Crysler Building - iconic NY. As is Grand Central Station and The Radio City Music Hall. Times Square isn’t that wonderful.
I also avoid Tourist restaurants such as Ellen’s. Broadway tickets are expensive so choose wisely. I go to the theatre a lot here so only choose the best. We tend to go to neighbourhood restaurants.
Another area worth exploring if you have time is SoHo. The streets are narrower and they have the iconic external fire escapes. Great little shops too.
Tiffany’s in 5th Avenue is fantastic to look inside. Also visit MoMa and the Guggenheim. These are world class. Going on the Staten Island ferry gets you great views of the Statue of Liberty. Go to Ellis Island to understand about immigration to NY.
The best advice is to think about what you enjoy. Buildings, museums, art galleries, iconic views, walks, culture, history, street scenes, theatre etc etc. The subway isn’t easy to use. You can end up somewhere you don’t expect to be. It’s grubby. Taxis are a lot easier. Also factor in tips and purchase tax. Tips are double the tax, so about 16% on top of the cost plus the tax. Prices are usually quoted without tax and restaurants likewise. Tipping is a way of life. You cannot avoid it so be prepared with smaller $ notes.