Over time you'll learn how to control the highlights but the shiny spots, as you call them, are specular highlights and actually should be there. You've controlled the highlights so there's no clipping and created some nice contrast between highlight and shadow.
Having a subject so that you don't have to do self portraits will help greatly so you can actually see how the face is being modeled.
Overall, for a first try you did very well.
I actually prefer the original shot to the one with the adjusted levels.
Work with the lights, move them around and experiment to see how the left,right, up and down position will affect the highlights, shadows and placement of the catchlights.
Think about composition as well. While there are no rules, per se, there are some rules of good portraiture that are great to know and apply especially at the beginning.
Having said that, you've got a bit too much negative space above your head. Based on the camera/lens orientation and composition that image would be better if cropped as an 8" x 10" but if you're going for an 8" x 12" (2:3) then you need to think more about the composition.