If you find you are unable to move a curve handle in all three axis, it is probably due to the curve type being set to 2D.

When you create the curve expand the Simple Curve dialog in the lower left of the viewport, or press F9:

From here choose 3D and then, if you want curved splines, choose Auto in the Output Curve Type.

The defaults are 2D and Vector and they may not be what you want for your curves.

The difference between Vector and Automatic are shown below:

You can also change from 2D to 3D after creating curves, if you forget initially. The setting is in the Properties panel on the Right (by default) of the Blender UI.

If you need to change the handle type from Vector to Automatic after creation, select the control points you want to change and press V then choose the type from the dialog: