It sounds like your goal is to be a Product Owner/Manager. I think your best career path is to start with Business Analyst or Project Coordinator types of roles at a SaaS company that builds products and is making profit. They will have Product Managers and likely some career options as you get more experience.

The difference with Product Manager vs. Project Manager I think is really a Product Manager will have to acquire or start with in-depth knowledge about the market a product operates in. In some ways a product manager might be a superset of skills including project management. There could be organizations big enough where these are different roles though. If you have knowledge/experience and interest in some domain, start there since that is helpful.

There are project mgmt classes and certifications like PMP or Scum Master. I don't think highly of orgs that require certifications on things like this, they tend to focus a little too much on strictly following a certain framework's guidelines as rules. They could be a way to get a job and some experience under your belt. I think the PMP cert is shorter if you have a college degree and it does require doing PM work as part of the cert, has different levels, etc.