Good news first - a designer is already a PM, a bit. Designers are driven by very much the same forces as PMs - find and solve customers' problems. Unsurprising that many designs are interested in switching to product management to have a greater influence on the overall product lifecycle. This switch, easier than from most other roles, still comes with several challenges, especially when it comes to business aspects of product management.
It's the viability of solutions where fresh designer PMs should focus first. Similarly to engineers, designers tend to be naturally biased towards the building, creating, solving. In most cases, that's a superpower but occasionally it might be a blind spot for designer PMs as some problems aren't worth being solved.
Leverage your strengths
Designers I worked with throughout the years were perfectly capable of supporting product development across the entire lifecycle. However, depending on the particular company and role, some designers are more focused on discovery activities and others on delivery. Both sets of skills are vital to become a successful PM. The techniques used in discovery such as user-centred design, jobs-to-be-done or a design sprint are bread and butter for a PM. But so as user interaction design, information architecture or prototyping in delivery. All those skills will be immediately applicable for a fresh designer PM and will fuel their transition to the new role.
Prototyping deserves a special mention as it's hard to overstate its importance in product management. You'd save a ton of time and piles of cash for your company if you use and even overuse prototypes. You can validate ideas and solutions with lightning speed and humble costs. Presenting your thoughts and communicating requirements become much easier with prototypes. You don't even need to write user stories if you have a good prototype. So if you're comfortable creating prototypes of any fidelity, by all means, exploit this superpower!
What to learn
Depending on where you're coming from as a designer, discovery or delivery, you might want to start with learning the missing bits. Even if your first PM role requires only one aspect, say a PM in discovery, you still need to learn the delivery part to become a well-rounded PM.
When you feel comfortable in both discovery and delivery, it's time to look into business aspects of product management. A successful product not only valuable, usable and feasible to create but also viable for your business. Hence in addition to asking what's in it for our customers, a PM should ask what's in it for us? Sure, not all products should bring money directly or right away. But eventually, your company needs to pay their bills which means your product needs to bring value to the business. Learning the financials that power their products and companies are essential for any PMs. To do that relatively quickly without completing a business degree - make some friends in the Finance department. There are folks in your company who know every decimal point after every penny that you spend or earn. Find them, offer them coffee with doughnuts and listen - they have countless stories to tell. To become successful PM you need to understand your costs, incomes, sales cycle, churn rate etc. You might use frameworks such as AAARR or Business model canvas to systemise this knowledge.
How to test drive
As a designer, chances are high you're already a part of a product team. Your experience will be very different depending on whether your team is empowered or not yet. Designers play a vital role in the empowered product teams and will experience pretty much the whole product management process alongside a PM and an engineer. Usually, this experience is enough for a designer to get a good idea of what PMs do and if it's the gig worth switching to. However, if you're not yet working in an empowered product team, or if product management is not well developed in your company, you might need to seek exposure to good PM practises. A good way to quickly do that is to join a side project alongside an experienced PM or take part in a product hackathon. Getting closer to a PM and working with her is usually a great way to understand better the role and decide if you want it. Without a doubt, any PM would love to have a designer by her side at any stage of the product lifecycle.
A word on a product designer role. It became a popular job title in recent years and might be a good preliminary step to a full PM gig. Product designer role might differ from company to company, but it roughly could be described as a designer making product decisions occasionally. The product designer will mostly be expected to contribute to delivery by doing UX-related work: information architecture, interaction and sometimes visual design as well. All those contributions can make or break a successful product and require countless decisions and intensive collaboration to make right. Hence a product designer role could be a perfect intermediate step for someone who wants a taste of product management without being responsible for the entire product lifecycle right away.
User researcher is another "in-between" role a designer who wants to become a PM might consider. The main responsibilities of a user researcher are in discovery, understanding customers and their problems. A successful user researcher is an expert in the problem domain and a champion of knowledge about the company's customers and users. To become a PM such a person would need to learn the delivery and the business side of building products.
Summary
The transition from a designer to PM could be easier than from other roles. A designer in an empowered product team would already be familiar with both discovery and delivery parts of product management. She could use her expert knowledge and techniques, such as prototyping, to supercharge the switch to a PM role. A fresh designer PM would need to focus her learning on the business aspects of creating successful products. Roles such as a product designer or a user researches might be useful intermediate steps on the way to a full PM job.