Changing jobs within an organization - talking to current supervisor?

I am looking for advice on telling my supervisor and department director that I plan on looking for a different job, ideally within my own organization. I'm unsure whether it is a good idea to tell them in advance at all, or how to do it, if so. Plus, I'm a child of immigrants, and a first-generation college grad and white collar professional as well. So am not sure what's even culturally normal for this sort of thing.

• I hope to stay within my own organization (municipal government), but move to a different department within my city. It's a large, coastal city in the USA with many different departments. I am open to jobs in other local city or county level governments, but want to stay within this sector and location. I like being a civil servant.
• I am hoping to go about this all upfront, and maybe even list current colleagues as references. I'd prefer that than apply and interview on the DL, and announce after I have a new job. But I know there are pros and cons to both methods.
• I have a positive, transparent relationship with my supervisor. I'm not worried about retaliation or getting pushed out, but rather, preserving a good relationship. And coming up with a continuity plan to get things in great shape for whomever replaces me. And leaving the door open in case an "ideal" job for me (job description + pay level) is someday created within my current department, a few years down the line.
• This is the first time I'm actively looking for a new job when things are going well, and I have some confidence. And it's also my first time actively applying for jobs within an organization where I already work. I'm mid-career. I'm not unemployed/financially desperate. I earn the median wage for where I live. I feel my current job is inherently worth doing/beneficial to society, rather than just a way to survive as an individual. I'm treated well at work and get along with my colleagues. So WHY mess with that?!
• Wanting a change is truly down to how I've grown/changed as both a person and a professional. AS A PROFESSIONAL: The job I want to be doing with my skills/experience simply does not exist in my department (in terms of day to day work and tasks, and salary). But some in other departments that are a good match for my skills/interests/pay goals have opened recently. And I am motivated to move to one of the larger departments where through a lateral move I will easily make tens of thousands of dollars more per year than I currently do, because I'm currently in a small department that our central HR has limited to lower compensation levels than other departments (our directors earn a lot less than the directors of other departments, etc...). My current job is very publicly prominent, which some would love, but I'm an introvert so this drains me and I would rather be behind the scenes most of the time. I don't use my graduate degree at all, and want a job where I do, both for intellectual contribution and pay reasons. AS A PERSON: logistics of my current job misalign with my life goals and responsibilities. My current job schedule is 40 hrs but not a normal M-F work week, and it's a profound conflict with partner/stepkids. I was single when I got the job 2 yrs ago, and now I'm not, so this has to change. I now have a family... and speaking of the education I'm not currently using to full impact, about $200k in student loans to pay off.

I recognize what an immense privilege it is to be in the position of wanting to move from one decent living wage job to an even "better" and more highly paid one. So I am hoping this conversation to be about growth, support, gratitude, and what's best for the public... just really anxious/unsure of how this all works. I would really appreciate anyones thought's/advice!

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