Do you just talk to your direct supervisior and lay out your career path wants or do you need to look for another position at some other shop or what's a working strategy here?
Camille Fournier's book, The Manager's Path, is a good read for learning about some of the common transitions from IC to manager to executive in software engineering.
You don't.
You build management experience.
1. Manage an intern.
2. Manage a junior's project.
3. Lead a three-person project.
4. Fill in for your manager's organizational work (planning meetings, higher level status syncs, etc) when they are on vacation.
Talk to your manager about doing these things, and why you want to do them. If the conversation doesn't go well, then you're in the wrong org to make it work.
Keep in mind that organizations will virtually always want to promote from within. The easiest way to show that you're capable of moving from Senior Eng to Staff Eng, or Senior Eng to Management is by meaningfully showing that you both understand what is expected in those roles, and that you're actively trying to incorporate those expectations into your every day work.
You should discuss the various roles / directions available to you with your manager, learn the expectations involved in those roles and work together to find opportunities that will help you move into those roles.
One thing, regardless of the direction you go in, that you can start working on right away is communication skills.
2. Many people will never become managers because they don't seem "managery." They don't dress right, they don't use the right buzzwords, they have a speech impediment, they speak too truthfully. There are all sorts of cultural factors too that people don't admit. For example, no manager would ever say this point 2 to you, because it's too honest and potentially upsetting.
3. A lot of people will only promote you to manager if they think you'll be able to make them look good, and that's the end of it.
Nobody is born a manager.
In my case, it was a sort of give and take in both directions. They needed a PM, and I had been doing some product work, mostly in the open source world, and wanted to give it a try professionally. They were explicit that being a first time PM was okay, and I was that I would be one.
I've been enjoying it a lot so far.
Disclosure: I'm a dev at Microsoft, on the Windows accessibility team.
If you don't let them know, once a manager role opens in your org, even if your name comes up as a potential person, anyone else who has asked for it and is being groomed to become a manager will probably get ahead of you. So, be that person instead.