Mammo, I'm in Suffolk, near Hampton Bays. Pleased to meet you!
I sure hope that I can help. Here are some things to consider. A good way to start might be:
1. Think first about what aspect of chickens are most important to your family. Some of us want really pretty chickens who are gorgeous eye-candy, so we might look for stunning silver-laced Wyandottes, Jubilee Orpingtons, a dramatic Pheonix longtailed pair. Some of us just want eggs, eggs, and more eggs-- as often as possible, and we might look at Golden Comets/Red sex-links or white Leghorns. Some of us want friendly, peaceful chickens that might be easily handled, so we might look for sweet Belgian D'anvers, chatty, sociable Speckled Sussex, or mellow Cochins. Also, some of us choose a chicken breed by egg color, we may want sky blue eggs and get Ameraucanas, green eggs and get Easter Eggers, or dark brown eggs and choose Welsumers or Marans. And, some of these breeds do multiple-duty, too, and can be nice egg producers and gorgeous and friendly and lay cool-colored eggs!

So, before buying any chickens with a limited space, decide what your family values most in chickens.
2. Next, a great place to look for started birds is right here on long island's Craigslist. Click "farm and garden," then type into your search box "chickens" or "hens" or "pullets." Better still, type in the
breed you're looking for, like "Orpington." After contacting the seller for more info, you would drive to the farm and check out the birds to see if they look healthy to you, and bringing a few covered cardboard boxes and a catch net (
Walmart sells them in the fishing dept), bring home the hens you see that you like.
Another option, more costly, is to order "started pullets" from hatcheries online. I'm not sure which ones have started pullets, but the websites would tell you. Examples of hatcheries are: Meyer, Murray McMurray, Ideal, Mt. Healthy, MyPetChicken, etc. This way, you can get exactly the breeds you want, at the ages you want, in the colors you want, be sure that none of them are roosters, plus, they'll likely be vaccinated against diseases like Marek's and maybe even bronchitis. Lastly, the forum has a Buy-Sell-Trade area. So, lots of options! Our girls came to us through Craigslist, and checking out a few local farms on the North Fork.
3. Cheap coop fill: In warm months, clean bagged playground sand from Lowe's or Home Depot for the floor, and a cheap bag of untreated white-colored pine shavings (for the nestboxes) from
Walmart's pet/hamster section is really affordable! In winter, using just shavings keeps the chickens warmer.
4. Feed sources: Agway is a good place to get a layer feed, and they have several types available. Chickens sometimes prefer crumbles to pellets, but they'll eat either. I bought a 40 lb bag of Nutrena Naturewise, a quality feed, for I think just around $14 or so, if I remember. (the Agway brand is cheaper, but they eat it faster and poop a lot more, so maybe there's less bioavailability/nutrition in it) There's also a feedstore in Eastport, but it's closed Sundays.
Anyway, I hope I've helped! Also, there's a lively, supportive, helpful forum chat thread here in the "Where am I! Where are you?" social section of the forum, called "Southern NY, Dutchess County and below" which you can join and connect with folks who have chickens and may live near you. Lots of Long Islanders participate! Some may have pullets ready to sell. Just a thought!