It's so very easy to visit Oaxaca on your own. When you are there, it is also very easy to book day trips to places including Monte Alban, the weekend markets in outlying towns, etc.
Sometimes you will find a driver and hire him for several trips during your stay..
The "big tree" take about two minutes to view..you will pass it on the way to other sites...
I'd say 5-6 days is perfect; I like to spend a week but I take things very slowly.
The center is very walkable.
You can take a taxi from the airport to your hotel. The taxi driver will probably ask you how long you are staying and where you want to go during your trip. If you like that person, you can arrange day trips right then.. If not, you can ask your hotel for the names of drivers that they like. It's all really easy....not at all daunting.
January/February are great months to visit the city.
You can fly from NYC to Mexico City and connect to a Oaxaca-bound flight. You could also take a bus from the capital; buses are very, very good in Mexico..MUCH better than those in the US.
I am happy to recommend hotels--there are many good ones in the center. Last time I was there, we stayed at the small CASA OAXACA, but there are many other great places to stay...it's a very good city for accommodations.
https://casaoaxaca.com.mx/en/index.php
It's a really delightful city and supremely easy to visit on your own--no organized tour needed...
If you want to combine city with beach time, there are many options in the state of Oaxaca, easy to reach from the capital, Oaxaca city.