How Can I solve this uneven mapping problem?
Not sure exactly what you mean (I am kinda noobish myself), but it looks like you might have some normals flipped. Try going into UV face select and see if you have some faces that are “backwards”.
The standard fix to try is
select the mesh
enter edit mode
Do AKey to select all the vertices
Mesh menu > Normals > Recalculate outside
Try a new render to see if it fixed the problem.
I fix the problem with uvwrap. Am I right to say that most of the time I well be using uvwrap even for a simple shape like this?
Could also be internal faces. Remove any faces inside your object - they will result in a non-manifold mesh and mess up the calculation of the normals.
No, I rarely use uvwrap for simple shapes. I just use the plane, sphere, cube, and cylinder MapTo options for that.
that what I use at first also. and the first image is what I get in mapping it that way.