I recently created some landscape code and added some diffuse lighting to the scene, however, to my disappointment, there are no shadows. I looked around the web for hours looki
No.
The easiest way I know of involves using a pregenerated shadow texture that is overlaid onto the terrain using multitexturing. The complicated part is generating this texture, but if you don't use directional lighting, a simple "big blurry dot" is usually better than nothing.
No. Rasterization is very bad at this (even recent AAA games have noticeable shadow artefacts), but everybody lives with it.
Solutions include (approx. from easiest/poorest to best/hardest) :
So the usual trick is to mix beautiful-but-static-only approaches with dynamic-but-not-that-good approaches. For instance, see my tutorials on lightmapping and shadowmapping.