What is the first Navy Enlisted/NCO Professional Development course, and what is generally taught there?

What happens is after you find out you are advanced to the next pay grade, your command will set up a time/day to do Petty Officer Indoc; the basis of how to learn to be a leader. After which, you are frocked (meaning you wear the rank but you will not be paid as of yet).
For E4-E6, we don't have a "school"; when you are an E7 or higher, you can elect to take a Leadership school (from my understanding). When you attain the rank of Petty Officer, your senior enlisted look differently at you. While you aren't doing grunt work majority of the time, they expect you to dish out the grunt work and deal with the complaining.

With counseling, are you talking about reprimanding or when we, as the "new" leader, needs assistance with a decision or to get an option from higher leadership? Since I've been a petty officer, my professional relationships with senior enlisted and officer has been very good; they have put a lot of trust into me and with what I can do with my job description. If I'm not able to assist, I do my best in researching a way for the task to be completed 150%.

Every promotion takes passing a course and exam and having appropriate marks on evals

When I served there was a basic Petty Officer Indoctrination, but nothing really to develop true leadership skills except for senior petty officers. I knew of a couple of E-5 that enrolled in the sergeants course on the Marine Corps base.

I made E4 and never received any training.

The course is officially referred to as the Petty Officer Selectee Leadership course. This is for Sailors advanced to E-4. There is also a course for E-5 through E-7. I have a copy of the facilitator material, let me know if this link doesn't work and we'll figure something else out.
https://www.dropbox.com/sh/j80bvl2wlwtv4yc/AAD_hYzH59WmVX4WW-gt74bBa?dl=0

Edit: These courses are command facilitated. Every command does it a little differently in how when they conduct the course. For example some may conduct the course prior to results posting and conduct the course for all eligible Sailors, while others may wait till after results are out and conduct it only for those that were advanced.

Unfortunately, the quality of the training is completely dependent on the command conducting it. They used to be actual courses outside of the command that were two weeks long and had much more sustenance and value.

I serve with 2 branches,the Navy for 23 and the Nat Guard for 8 and I recently retired, As for the Navy back in 83 when I made E4, we had to go to POIC,Petty Officer Ind Class for 1 week of training. It taught us a lot on leadership,professionalism and respect of being a PO, but it doesn't stop there. Each step that you make in the ranks,shows that you earn that spot and respect to teach others who are coming behind you so that they will pass it along to the next sailor thats willing to make that goal.

It changed a lot and had various iterations from 1983 to 2012, when I retired as an MCPO (E9). When you picked up PO3 (E4), you went through a command-facilitated PO Indoc, which was usually week long and usually led by a group of PO1's (E6) from various rates. In addition to basic leadership, new petty officers will discuss values, discipline and Navy standards. The course was developed to educate Sailors in a variety of leadership-related subjects that will help build a foundation for future self-development.

Good luck. My son has been Navy for about 10 years, testing for CPO next month, and I'm not sure I still understand it. The big difference in both Officers and Enlisted is that their promotion system is not school based like the Army, so apparently the most important thing is to get all your "Quals" for your job and as much as you can for the next rank, then your evals and test scores determine if you go before the Board.
Most of the education system after A-School seems to be OJT, which also seems to vary in effectiveness between commands and commanders.

The Coast Guard is essentially built like the Navy with some exceptions. The E-4’s get a small taste of professional development during A-school (a.k.a. AIT). CG members must be an E-3 (SN/FN) to attend A-school. When they graduate they’re automatically promoted to E-4 (PO3). These members will not be allow to test for E-5 (PO2) until they’ve attended a Leadership and Management School (LAMS) and completed other job and military competencies. When E-6’s (PO1) are notified they’re "above the cut" for E-7 (CPO), they’re automatically assigned a Chief Petty Officer’s Academy class attendance date. This is a mandatory 6-week resident course held in Petaluma, CA. Nearly all these members are frocked (wearing the rank but receiving none of the pay) on the day before their class attendance date. They start receiving the pay upon their official advancement date unless other circumstances arises (i.e. disciplinary). Senior Chiefs (E-8) must attend the Command Senior Enlisted Leadership (CSEL) course and be screened prior to being selected for a Command Cadre position. There’s only a handful of Command Senior Chief billets in the fleet. Most are at the E-9 (MCPO) rank for Command Master Chief. (Not all E-9s are Command Master Chiefs. Some aviation Master Chiefs are billeted as Leading Chief Petty Officers (LCPOs) at the biggest air stations mostly.) Some Senior Chiefs and Master Chiefs compete to attend the Army’s Sergeant Major’s Academy in addition to all of the above. These members are usually checking the block to get as much professional development as possible for Gold Badge selection. Gold Badge Command Master Chiefs serves Area and District Commands (VADM-O9 and RADM-O8 level commands) and are more favorably considered for selection to Master Chief Petty Officer of the Coast Guard. Although, all E-9’s are solicited and any E-9 can apply to MCPO-CG.

Its really up to your command to send you to Indoc. If you are in a mission essential billet they can forego sending you. I advanced to PO3 directly out of Rescue Swimmer school and PO2 while on det. But all Chiefs go to indoc for sure.