OP, (not a fan of your user ID so it will be OP)
Are you good to go regarding hunter safety and bow safety certification. (I assume so.)
As a Nassau county resident, you can get a leisure pass for about 30 or 35 bucks and sight in your shotgun at 29 yards in the indoor range at Mitchel Field in Uniondale, just down the road from you.
That same leisure pass will allow you unrestricted (sometimes they check for leisure passes on weekends) at Cedar Creek Park, off Merrick Road on the Seaford/Wantagh border. There is a BYO archery field opposite the children's playground (yes, you read that right, this is New York after all where little makes sense at times.) The field is bordered by a practically see through wood stockade fence but goes well over 150 yards to a berm, where you will be shooting in a southerly direction. High winds can be a problem. Weekends there can be a zoo as ignorant parents have taken their Hunger Games descendants and while standing behind them let the little rug rats run to get all three of their arrows without waiting for a "CLEAR" call from other archers. No broadheads, and, I believe, no crossbows allowed. Saturday mornings will also see a group of Nassau Bowmen (google them) sometimes in a tournament. They have a small locked shack just N, of the range. There are only three picnic tables to lay your gear on inside the fence so be prepared to lay your stuff on the ground. Speaking of stuff, its strictly BYO. No targets, rental equipment, and most importantly no range officer. Park just north of the children's playground. Only on nice days is that lot full. With regard to where to buy a bow, I was quite happy 7 or 8 years ago when I went to Smith Point Archery in Patchogue on Montauk Highway a block west of 112. I will attach my admittedly biased review below. If you are, tongue in cheek, working from home, try Cedar Creek on a weekday first.
Regarding Suffolk hunting, I am now on a nuisance permit but this may be the first year I do the shotgun lottery in January after a four year hiatus. Do not let the term lottery discourage you ! While its possible that if you get a very high number in the drawing you would not be able to get a spot and a date for two weeks into the January season, I have rarely gotten a date later than January 11. One year I got a number like 53, selected my date and spot and a month later experienced zero degrees plus a stiff wind. For the first time this year, because of the china virus, you apply online, not sure if that will last for the 21-22 season. Anyway, besides being "stuck" with a single spot that you chose, on a date that you also selected a month earlier, you can also appear out at the Ridge DEC check station and go "standby." There are always a few spots available since people have stuff come up and can not hunt on the date they selected weeks earlier. You will need one of the pieces of paper from your original shotgun hunt to go standby, which back in the early 2000s I did several times. For bow hunting on Suffolk state properties I will hope that someone else chimes in. My knowledge is old and useless from what I can see on the website now.
"I am much closer to C & B (NOW CLOSED)and before his passing my buddy worked there quite a bit, being the then President of Suffolk Archers.
I went to Smith Point. You will get mixed reviews re: C and B, very few negatives regarding Smith Point. I posted the following over a year ago.
Went last night for the lesson provided gratis upon purchase of a new bow. I had previously exchanged some informative emails with Jared there, (who replies within a work day) and ten days ago spent a long time selecting my first compound with Craig. My
friend and hunting partner, who helps DEC Hunter Safety provide some of the certificate courses at Smith Point Archery, and I, had discussed what I was looking for in a bow, and presented this rough list to Craig when I showed up. While Craig could have gone to a more expensive line, he instead felt that the more adaptable bow, keeping in mind my parameters and preferences, was probably the Martin Cougar package. ( I did not express any manufacturers preference and by mere chance, which no one knew, my recurve is a Martin.) Craig explained the various upgrades that I could and should consider, some of which I selected such as tool-less adjustable sights, a better stabilizer, an adjustable release, etc. Again, this was dealing with a fellow archer and not a salesman, and hopefully the owner of Smith Point Archery, if he sees this, will understand my intent and compliment. Although I showed up on an afternoon during bowhunting season, I went off for a Carvel next door, while Dave the mechanic immediately went to work setting up my bow. Upon returning, still another archer there, Tim, set me up at the large indoor range where I shot my first arrows from a compound bow. Getting back to last night's lesson, it was practically a private hour plus lesson as Craig had just myself and one other gentleman in the classroom/range, while Craig's wife assisted a young lady with her own lesson. I fired away until my shoulder was thoroughly fatigued, lol, and not due to any time constraints, all the while being advised by Craig with release, follow through, grip, etc. I can definitely see availing myself of the free range time also provided upon purchase, of a year's duration, this winter trekking out to Patchogue. (for those interested or influenced, its 200 yards west of 112 on East Main St.) I might mention that living in Nassau there were a couple of closer shops I could journey to, but unlike them, I had never heard a single negative about Smith Point."