The dark regions are caused by inverted normals. This is a thing many people struggle with when starting to make 3D art. Blender has a function for quickly fixing most of such cases - all you need to do is (from within Edit Mode) choose Mesh > Normals > Recalculate Outside (or Ctrl+N).
The ugly "pinching" effect is a result of all the subdivided geometry pulling in toward one point, creating a spoke-like look. This can be fixed by making an all-quad mesh instead. Try selecting that center vertex, deleting it, and instead select the remaining edges and doing a Grid Fill. Tweak the Grid Fill settings until they are to your liking.
But wait... you can't do that just yet, because there is one more problem with your mesh: you're trying to create boundaries by filling in faces to section off areas of it, but this will not work. Making paper-thin hymen-like membranous faces is a topological no-no. This results in what is known as a non-manifold mesh. To explain what does or does not encompass the definition of "non-manifold" is beyond the scope of this answer, but to sum it up, understanding that concept will enable you to create geometry that will result in the shading you want to achieve.
Your wine glass: fixed (Screen Shot - The lighting is crap, but I think it illustrates the point well enough)
Your wine glass: fixed (.blend)
And here is a side profile illustrating the boundaries of the areas you want to have thickness:
For comparison, here is the geometry you had created (after the Solidify Modifier has been applied):
I've outlined the two problem areas with the Grease Pencil tool. To sum it up, you don't want all of the wine glass to be "solidified". Think about the structure of a wine glass - the foot and stem of the glass are not hollow inside, rather they are solid glass. Yet by adding a Solidify Modifier to the entire wine glass that is what you have made - a wine glass with a hollow bottom. This is where you are having difficulty, because you haven't created geometry that accurately reflects the structure of a wine glass.
The second problematic area (which is related to the first), is that to compensate for the hollow area in the stem you tried to just add some faces between the stem and the cup (what I referred to as the "membrane"). This is where you created non-manifold geometry and further confused the problem.
Understanding these concepts will help you in every 3D sculpting project you undertake from now on, so I can't stress enough how important this is.
Here's a mediocre Maya tutorial explaining non-manifold geometry: https://www.youtube.com/watch?v=vrqxp89ilM4&spfreload=10 (sorry, it's the best video explanation I could find - of course this concept is the same whether using Maya, Max, Blender, etc.). Hopefully this will help you out beginning your understanding of non-manifold vs manifold geometry. In Blender you can highlight non-manifold geometry by pressing Shift+Ctrl+Alt+M. You can also select non-man geo by going to the menu within Edit Mode and choosing Select > Non-Manifold.