What ideas or projects would you try to take on and lead after landing your first product management role?

If you are an entry-level product manager or a first-time PM switching from a different function, executing a feature end-to-end is the best place to start. During execution, one can learn system behavior and develop necessary POVs, such as user behavior and the jobs-to-be-done for the persona. During this process, a PM works closely with peers in the User Experience and Engineering teams, helping the PM understand how the team operates. Also, during interactions with customers and the sales team (B2B) at different product/feature launch stages, a PM can demonstrate their expertise in the feature and the product. Conversations with the sales team and customers are where product strategies and future roadmaps usually develop,  outside of top-down, market-driven strategies handed down by leadership teams.

Finally, starting with execution does not mean the strategy is the wrong place to start. In large companies, strategies are often vague and require cross-functional stakeholder approval, and it's a bit harder to establish your POV with members outside your team.

The job of PM relies on influence, not on ideas. As a young PM, I always suggest to focus on building influence small steps by small steps. Get small wins. People start to follow you because they see that you can achieve things, not because of your ideas. So as a PM start by achieving things, pick one thing and get it done in a short amount of time. And repeat it.