Every company is different but I can tell you what I've been working on as a part of my transition:
I have two tips for breaking into Product Management but I'm sure there are many more.
The first is to work with Product Management in your current role and/or talk to current Product Managers. Product Management can be seen as a very appealing job but many don't understand the day to day and trade offs of the role. By working with or informationally interviewing folks who are currently doing the job, you'll get more information on wether the job is actually for you and what appeals to you about it.
The second is to ask your manager if you can work with Product or use Product principles in building something new or improving an existing thing as a special project. This could be an internal process, application, or something else. The idea is to apply product thinking and process to something in your current job to see if you like the way of doing work and the type of work.
Both of these suggestions will help you to both understand if Product is a career for you but also will prepare you to interview for the role.
Product management is a combination of a lot of different knowledge and skillsets but the core of the job in my opinion is three primary things: curiousity with a maniacle focus on the customer and getting sh*t done. Therefore, I'm really looking for folks who have shown by finding problems, understanding them with a lot of empathy, and solving them proactively and showing ugency and results orientation in driving outcomes (blasting through roadblocks and showing measurable outcomes). In my experience, these can be demonstrated in most roles - as a support engineer or CSM helping customers to get to outcomes, as an analyst building analysis to drive decisions or tools to empower teammates, as a program manager coordinating a team to get to an outcome, or as an engineer helping to guide the best solution. Demonstrating these three things both in how you write about your experience in your current role on your resume and how you interview is a sure way to get my attention.
Based on my experience working at both smaller startups (25-50 people) and larger companies (1000+) as a product manager, there is merrit to both. I really appreciated starting my product career at a larger company because there was established process / best practices and resources like design and dedicated engineering teams and other practitioners to learn from. This allowed for me to understand the mindset and process of doing product at scale which I then was able to take with me when I moved into working at startups, helping to evolve the startup chaos to create industry standard processes for product, design, engineering and stakeholders. However, the pace at the larger companies I've worked at has been slower and therefore it takes a while to get the cycles in and cycles are what allow you to refine your product process and sense. Working at a startup on the other hand offers lots of cycles, which really allows for quick practice, lots of mistakes, and results in acceleration in learning. I don't think you can go wrong either way but it definitely depends on the amount of structure vs. chaos that you're interested in being a part of as you're developing.
I think the most common mistake that I see is jumping to solutions quickly. This is definitely my biggest mistake as I started as a product manager. For me, it was/is really easy to think I know the answer and to move quickly from the unknown/undefined to the known/defined stage and ultimately check something off the list by delivering it into the hands of customers. However, the role of the product manager is to stay in curiousity and research and the unknown for long enough in order to get enough information from customers and then from your crossfunctional partners to define to the best solution to solve the customer's pain while balancing that with the goals of the organization and your core product principles. And all this while driving urgency around achieving results. A tall order!