I'll recommend you this:
Start with a flat geometry (in your case you can delete the extruded faces):
Notice that I only add one circle (You should lower the circle vertices number for ease of work). In this particular case the two big figures (the big circle and the ellipse) have 32 vertices and the small one have 16 vertices.
Select 3 quarters of your geometry like this:
Erase the vertices
Then add a mirror modifier that affect the X and Y axis to get this result:
Now fill up the geometry the way you need it:
If you get this kind of effect, the darker faces (it may not happen) this is due to some normals that have been gone the wrong way, to fix this select your entire mesh by pressing A and then press Crtl + N to flip normals.
Now, get a better view angle.
Extrude on the Z axis, press e (extrude) and move your mouse until you get the desired results.
For more detailed mesh you can add two modifiers. First the Edge Modifier and then the Subsurf modifier. The order of the modifiers will impact in how your mesh looks.
Maybe it would be easier if you did it another way (destructive version):
- Create a circle with 3 inner circular edge loops.
- Create your central oval, extrude it down.
- Cut off your mesh, keep only 1/4th and give it 2 mirrors, one on the X axis, the other on the Y axis.
- Create a hole, extrude once, and use W > LoopTools > Circle to make your hole circular.
- Extrude down your circle.
- Extrude up the whole mesh to give it thickness.
- Extrude up the central hole.
- Give some extra edge loops so the angles will be sharp when you'll add a Subdivision Surface modifier.