First you really need to have an american number or at least be able to call with your phone in NYC, it is almost necessary when visiting flats. I couldn’t call people for my first apartment visits and it was really hard for me to contact them and be sure that they were in their apartment at the right time.
I finally bought a j1 sim card. You could go and check there https://www.j1simcards.com/, I think the offer is quite interesting for international people: you pay 40$/month and you have unlimited text and calls in more than 70 countries, and have 2GB of internet.
For finding a flat you can look on Craiglist. There are a lot of posts, but personally I don’t really like this website. I think it isn’t very agreable to look at and I didn’t received many answers. So send many messages to different people if you want to get some answers and plan some visits.
You can also try to look on some Facebook groups for housing in NYC, there are plenty of them! I found that the french group “Les frenchy à NY” was very helpful. There are also: NYC housing helper, New York city room and roomates, Gypsy housing NYC… In my opinion, the good thing about looking on facebook groups, is that you can usually check who posted the ad.
When you are visiting apartments, don’t be afraid to look everywhere! Open kitchen cupboards and fridge to check that they are not dirty or that there are no cockroaches sleeping inside (in NYC there are plenty of them, and I’m sure you don’t want to have your first breakfast facing a cockroach!). If there are roommates in the apartment you can ask them where you will be able to store all your food and stuff if you stay there. You can also prepare a list of questions before visiting so you will be sure to ask everything you need to know.
Prepare yourself to give an answer pretty quickly. There are many people in NYC like you looking for a flat so it often works on a first-come first-served basis. Also be sure to have enough money: usually landlords ask for at least one month rent + the deposit (often same amount as one month rent).
There are many apartments in NYC, but they are often very expensive. So maybe you will have to search for a little time before finding a flat where you can live in. You will certainly have to make some compromises: either a flat with nice roomates but no AC, or one very comfortable but unfriendly roomates, no windows in the common areas but as sunny bedroom, or an incredible living room and a little bedroom with no window…
I hope these few tips will be helpful, good luck in your apartment research!