What is the best way to get started trying to reenlist to go AD after being discharged from the NY Army NG in 2017 with an RE-3 code?

As CPT Lawrence Cable stated, talk to a recruiter. That will always be the source of the most accurate answer if you're exploring reenlisting.

Specific to your situation, on your NGB 22 you should have an entry in block 26 (Reenlistment Eligibility) next to the RE code which is the separation code. Refer to NGB Regulation 600-200* para 6-35 to match up the details of your separation code.

Why? Because all RE-3 code are not the same and the waiver will depend on the separation code. For example, even though someone was separated for misconduct and given a RE 3 code, there are certain types of misconduct that DoD will not allow the Services to give a waiver for.
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* NGR 600-200 (Enlisted Personnel Management) - https://www.ngbpmc.ng.mil/Portals/27/Publications/NGR/NGR%20600-200_20210325_v2.pdf
* Army Directive 2020-09 (Appointment and Enlistment Waivers) - https://recruiting.army.mil/Portals/15/ARN30547-ARMY_DIR_2020-09-000-WEB-1.pdf
* USAREC Memo - Waivers Submission Instructions and Document Requirements for Army
Directive (AD) 2020-09 - https://recruiting.army.mil/Portals/15/Website%20AD%202020-09%20Memo.pdf

Talk to an Army Recruiter?
R3 means you require a waiver. What you need a waiver for will determine how easy or impossible it will be to re-enlist. The is also a Separation Code on line 26 of a DD214, don't remember on an NGB22, that will give reason for the R3.

Again, talk to a recruiter. They will know what they can and cannot get waivered.