How can I add thickness to a curve? I already tried with the extrude and depth options in the curve settings, but it doesn't work properly.
All you need to do is set the Fill to Full on the curve to get a complete 3D shape. It defaults to Half, which is why you only have half of you shape showing.
Add a nurbs circle and use it as bevel object:
You can control the thickness by scaling the circle.
The simplest way to do this is to use a Solidify modifier.
This is the result from the default Bezier curve.