How do I make realistic shadows (that not not are completely black)?

You could try out Buttons Window/Ambient Occlusion found in Shading/Amb Occ. Check out manual pages too:

Also, if you don’t want the long render time associated with AO (mentioned above) just put in a couple back lights with low intensity to lighten everything a bit. It’s not as good as AO, but it will look lots better if done right and save lots o rendering time.

If I use Ambient occlusion the render get corny. Is there any way to avoid that?

Increase the samples (which of course increases rendertime as well).

You could do your shadows as a separate pass, and composite them in with whatever degree of opaqueness you then desire.

You could use a “shadow only” spotlight to provide shadows.

You could use a lamp that is set to only illuminate the layer upon which the shadows will fall.

You could cause the material upon which the shadows fall to emit light (or to reflect more ambient-light).

What you are really trying to do is to decrease the contrast in the shadowed area. You’d like to do it without adding more lights. You don’t necessarily have to do that in the same way that you might do on a real photo stage.

Decided that Ambient Occlusion is the easiest to use and seems to be the best so I use that. But I have a little problem. The renders with Ambient Occlusion is lighter than the one without. Is it possible to fix this? And How?

Yes, us the SUB feature in AO to darken only.