I decided simple value noise (essentially just random values) with Fractional Brownian Motion (fBm) would be enough for now. Besides, with my implementation it will be easy to expand later to Perlin, Simplex or some other type of noise. Natalya Tatarchuk’s talk that I already linked previously has an excellent explanation of fBm when applied to noise. I already have vague plans in mind for more complex multi-stage terrain generation with terrain types, maybe some erosion etc, but I'll have to concentrate on more important stuff first. Like gameplay. But first, the pictures I promised, click for larger versions. And don't mind the ugly terrain texture.

Value noise (with linear interpolation) on its own isn't particularly spectacular:

With quintic interpolation (6t⁵ - 15t⁴ + 10t³), as used in Simplex noise, it looks much better, but it's still quite boring:

Combining several frequencies of noise with fBm immediately generates much more interesting results. Four octaves of noise is enough to start showing a difference:

Increasing octaves to eight provides more interesting details:

Increasing octaves further doesn't really add anything you can see in this application. This is a picture of terrain with 16 octaves of noise: