What regulations would you suggest a NCO to really dive into and learn, upon taking his/her first role as a leader with soldiers below them?

AR 600-20, 350-1 and 670-1

Regulations (not ADPs, TMs etc) Knowing what is in them and being able to find things.
600-20 - especially family care plans
350-1
600-9
670-1
735-5
608-99 Pre-Separation Support
635-200 Enlisted Separations

In the spirit of Sergeant Morales and my old PSG Tom Torres, I wish every first line would learn where to find info on the following to help their Joes:
- BAH
- how finance dysfunctions
- attends the Garrison R2 program overview so they can understand what help is out there for them and their soldiers.
- how to conduct PT without hurting people.

Your unit SOP and Commanders intent

They should learn as much as they can about counseling and awards. That’s something they doesn’t necessarily require a PDF unless to pin point a specific question or clarification. Soldiers want to know how they’re doing but it needs to be documented to preserve the integrity of the Army.

Also have him read the Army Leader Development Steategy. It's just like 25 pages. Quick read.

First thing I'll say is that my previous unit, you were assigned as a team leader while being a Specialist before they allowed you to pursue becoming an NCO which i definitely agree with. Put you in the hot seat to see how you take on the role.

This said, as an E5 my leadership style was mission first and after that I explained to my soldiers, don't do anything stupid to end up in jail or to land you in hell, after that I'll see you at the next hit time. I never micro managed. Not my style because i trust my soldiers to be adults and to motivate themselves. If they failed then i would take control and give the tools to get it done. After which I let you go again with enough rope to hang yourself.

My Staff Sergeant was the same way but he was more by the books than i was. He allowed you to be human and left you alone for the most part but you slipped once you got smoked and paper trail.

I refused to do paperwork on my guys. I told my soldiers that paperwork is just words but you can be a hell of a soldier, kick ass on the battlefield, but still be a shitty ass person.

So I'd say, teach NCOs to learn about their soldiers. Who they are, their likes, their dislikes, their family, but also remember that you came from the bottom too. Empathy and trust goes a long way.

Send them to the Basic Leader Course (BLC) if not already. BLC will teach the baselines of everything leadership and teaching skill level 1 tasks. It's on the young leader and the supervisors to progress that BLC-taught knowledge. For you, set the perfect example through your actions and spend lots of quality, two-way conversation with that leader to discuss his or her dilemmas, personal and professional challenges, and how you are going to help arrives them to their short and long-term goals. One bonus tip is that everything in the Basic Leader Course is pre-packaged to turn around and teach your skill level 1 subordinates, and this is why former BLC instructors are so valuable to the force.

There are some career progression maps that will show what publications we should be familiar with at what rank. Think it was the DA PAM 600-25 on the milsuite site.
I also recommend AR 27-10, so they can advise their people on UCMJ stuff, and AR 735-5 on how to maintain their property properly as a squad leader.

Ar 385-10 and AR 600–55 (yes even if you’re not an 88m you should get familiar with it)

Read books on some great leaders, learn what they did best and try to emulate them. Also, look at the bad, or what wouldn't fly today and make sure your don't do that.