Vulkan represents a low-level way of talking to the GPU more or less on the GPU's terms. As opposed to OpenGL which is a way of talking to the GPU on boomer CAD programmers' terms.
As for whether Vulkan is going to replace OpenGL... yes and no. Over the next 10 or so years we may see OpenGL drivers for GPUs get replaced with Vulkan drivers plus an OpenGL layer based on Vulkan. But you will still be able to develop to OpenGL.
Ad for what you should start with... what do you plan on doing? If you're looking to write a game, use a widely available engine like Unreal or Godot that abstracts the graphics library details. If you're looking to write your own engine... seriously, seriously reconsider.
If you're still interested in programming immediate-mode graphics below the engine layer, start with DirectX 11 and 12. They're much more tractable than OpenGL and Vulkan and have unbeatable tooling support from Microsoft. Even if you're targeting Linux, you will have much more binary compatibility across distros by shipping it as a Windows program and running it under Wine.
Regarding the last statement, segregation of game and engine is one way of thinking. But many a times, you do not need a full fledged map editor, sculptor etc. You just implement the parts you need for your game.
That still needs to be seriously, seriously reconsidered in favor of basing your game on a major engine. There's just so much tooling and workflow around the big engines. Personnel to help will also be easier to obtain.
"Vulkan represents a low-level way of talking to the GPU more or less on the GPU's terms. As opposed to OpenGL which is a way of talking to the GPU on boomer CAD programmers' terms."
Kind of like that description. I've been doing GL since before it was OpenGL.
In my Common Lisp 3D system [1], I've been happily using very simple immediate mode OpenGL calls, taking advantage of CAD-like features such as drawing wireframes of various thickness over shaded polygons. OpenGL provides specific Z-buffer API calls to facilitate such things.
Is there a simple way of doing such CAD-centric things in Vulkan?
As for whether Vulkan is going to replace OpenGL... yes and no. Over the next 10 or so years we may see OpenGL drivers for GPUs get replaced with Vulkan drivers plus an OpenGL layer based on Vulkan. But you will still be able to develop to OpenGL.
Ad for what you should start with... what do you plan on doing? If you're looking to write a game, use a widely available engine like Unreal or Godot that abstracts the graphics library details. If you're looking to write your own engine... seriously, seriously reconsider.
If you're still interested in programming immediate-mode graphics below the engine layer, start with DirectX 11 and 12. They're much more tractable than OpenGL and Vulkan and have unbeatable tooling support from Microsoft. Even if you're targeting Linux, you will have much more binary compatibility across distros by shipping it as a Windows program and running it under Wine.