Having a computer engineering degree helps. Outside of the embedded equivalent of CRUD, you'll have to know the details of how a computer works. How memory visibility and synchronization works, what mmio is, basic assembly, interrupt concepts, etc. I'd actually recommend reading a book on operating systems. A lot of embedded devices that are pure compute require knowledge of operating systems because you're basically building software to enable a user to interact with the accelerator or sensor on the device.

I'd also learn a tool like yocto or buildroot, both tools for creating Linux images. They're not great tools, but they are widely used in creating embedded Linux images.

If you want to solve domain specific issues (like getting into DSP) or doing high speed networking, you have to find an entry level job in that field and have the requisite background. It's hard to do DSP without an EE background for example.