How can I do cyclist as a professional sport and where in NYC?

Hang on Christine...  You're about to get a lesson about both grammar & cycling.  The way it's worded, you want "to do" a professional cyclist - as in jump his bones.  Or maybe you prefer a female pro cyclist.  ???  Whatever turns you on, makes no difference to me.  Try it this way...  "How can I learn about professional cycling in NYC?"  See the difference?   

Next...nobody and I do mean NOBODY starts as a "professional cyclist".  Pro cyclists all belong to a team.  You don't choose a team.  They choose you IF (big IF) you're good enough.  Here's a list of names from one team in Great Britain - https://www.uci.org/road/teams/TeamDetail/13993/10...

Just like any other sport, you start as an amateur.  If still a youngster, you start in the Junior Division of USA Cycling.  https://usacycling.org/  I might be wrong, but I believe once above the age of 16 you can start in CAT 5 cycling (category 5).  That's the lowest rank.  Build up enough points & you can move up to CAT 4, then CAT 3, etc.  If and when you ever reach CAT 1 racing, a pro team MIGHT look at you.  Here's a list of events in the New York State area.  https://usacycling.org/events-search?discipline=19...  If New York is anything like the Midwest, many races have been cancelled due to COVID-19.  

I don't know of any "Professional Road Races" in or around NYC.  The big 3 are the Tour de France, The Giro d’Italia (Tour of Italy) and the Tour of California which is on a "hiatus" this year according to the organizers. 

https://www.bicycling.com/news/a29640173/amgen-tou...

Now...another question for you.  Did you mean professional ROAD racing or mountain bike racing?  There's more than one type of pro races.  There's also CX (cyclocross) races.  Posted Jul 28, 2020, Races You Can’t Miss in Late 2020 from Bicycling Magazine.  https://www.bicycling.com/racing/a33444818/pro-cyc... 

It's not a "professional race".  But there is one in NYC for people with folding Brompton bikes. 🤣

https://youtu.be/IrNLP3QMDpE   

The first thing to consider is your age. The 30+ pro cyclist are nearing retirement and have been in racing for 20+ years. Go to some reputable bicycle shops and they can probably tell you some good amateur events you can enter. If you don't want serious injuries from competing in these amateur events get a road bike. That folding bike won't stand a chance and will do major damage to your body. 

Update: 

Didn't even realize there was such an event (folding bike race). RB has probably given you the best help as far as your folding bike is concerned. Such an event is setup based on the limitations of the bike similar to nostalgia races. Vs junior or senior races which are based on the limitations of the riders.