There is a fork in the PM career path road: one is becoming a people manager, the other becoming an expert in a deep thinking product area sans managing a team.
My recommendation is to figure out which one is right for you. Many folks want to jump into management simply because they think this is the only way to grow, make more $$$ and so on. That is not true. Big and small orgs I have been a part of value senior individual contributors that are passionate about their individual craft. Speak with folks from both paths, and see which one resonates more with you. Try mentoring people and see if you like helping others succeed through your guidance as a "management path" check.
Then, share your thinking with your manager to get them to help you moving along this path.
This is a very interesting question and one that I keep touching upon on almost all career conversations with my teams & mentees. There is no one typical career path for PMs, which can be both liberating and challenging at the same time!
It's liberating since PMs have the chance to shape their careers to what they would like it to be, playing to their strengths and having a fulfilling life & career. It's challenging since every industry and many firms in the same industry have different definitions & requirements for what a PM is! For instance, even within FAANG firms, the definition of a PM is different - Google requires PMs to market the work of their Engineers, Apple PMs are usually good at their domain of expertise but execute on Sr Management decisions, Amazon PMs are Business/ Program Managers with heightened focus on a few metrics, Netflix PMs are very Technical (most of their PMs were former Engineers or Architects) and Meta offers almost the full buffet but also the most agency & empowerment to PMs. I touched upon some of the PM archetypes in another answer and its great to see that PMs can succeed in any shape or form tht adds value to their firm!
PMs who are early in their careers will do well to join established firms to understand the PM tracks and how senior PMs have shaped their careers. Working in a start-up or a 0-1 environment usually turbo-charges a PM's career but only if the PM is aware of their strengths & know-how to leverage them. I have found that alternating between large firms & startups or between established products/ projects and 0-1 initiatives is the best way one can gain the most perspective and shape their careers as PMs.
There are different paths that each product manager takes, but the common ones I've seen are:
1. Joining a tech company as an Associate PM or an intern straight from college.
For college grads, I suggest starting by connecting with other product managers (e.g. via LinkedIn) to better understand what we do. There are great books available on this topic as well -- "Cracking PM Interview" is among my favorites. I also created a series of videos explaining tech jobs and what do I do in more detail - https://www.youtube.com/channel/UCsAz_arwNkiPobhi09VrMFg
2. Transition from other roles e.g. Engineering, Professional Services, Support.
This path is easier, as it assumes that you are already in a tech company and can make connections with internal PMs. Picking a PM as a mentor or just becoming a friend with one is a great place to start. I also need to point out that PMs sit at the intersection of Business, Technology, and UX (Customer) -- that is why engineers who transition to a PM team will have an advantage as they understand the technology much deeper. On the other hand, someone in Support who wants to become a PM brings a much deeper understanding of a customer.