I've been using blender 2.8 and it was fine following along youtube videos and the previous exercises here, however the settings used here are very different in 2.8 so I couldn't even paint properly. And I couldn't import my unwrapped axe file from 2.8 to 2.79 so I had to unwrap again in my new 2.79 file.
Until the 2.8 version of this exercise comes out, stick with your old blender version!
It can be completed in Blender 2.8 and here's how.
Make sure to create a new images texture in the UV drop down box and name it paint or whatever you like.
Go to the material tab and add a new material node make sure to change the Base Color option in the node to image texture.
Now in the drop down under the node for the Base Color select your image texture that you created (I called mine paint) or whatever name you gave your image when you created it in the UV area.
Now go back to Texture Paint in blender 2.8 and you can follow the tutorials as normal.