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Vulkan is pretty boring to be honest. I'm very doubtful that you want to bother with it if you're just starting. Using these APIs is like filling out tax forms, but Vulkan is a lot more effort than OpenGL (but, OpenGL is weird...) Learning shaders in OpenGL will be pretty transferable knowledge. Vulkan uses a lower-level assembly-like language, SPIR-V that doesn't pretend to be C (like GLSL) but there are GLSL to SPIR-V compilers. I can't comment on DX (I haven't used it.)
Here's a tutorial on drawing you first triangle: https://software.intel.com/en-us/articles/api-without-secret... . Note that there is a lottttt of code to do this (this is part 3...) I really think you could learn more about modern graphics with glBegin(GL_TRIANGLES) and writing GLSL shaders than boring yourself with Vulkan.
If you do want to play with Vulkan there is this book (piggy-backing off the rep from the old "OpenGL Bible") https://www.amazon.com/Vulkan-Programming-Guide-Official-Lea... . I can't give it an honest review because I got bored; I plan to pick it up again later...