Learn Turkish: Güneş sayesinde dağlar belirdi. - Thanks to the sun, the mountains appeared.

Both are actually correct! Sayesinde means "thanks to" (literally "in the shadow of"). You can use it without the genitive suffix to make a general compound (Güneş sayesinde), or with the genitive case (Güneş'in sayesinde) to be more specific. Note that with pronouns, you always need the genitive (like benim sayemde - thanks to me).

How would I use this lesson's grammar to say the mountains gradually appeared?

You can add gittikçe or giderek right before the verb. Saying Güneş sayesinde dağlar giderek belirdi means "Thanks to the sun, the mountains gradually appeared." It changes the sudden action of appearing into a slow, step-by-step emergence.

Why use belirdi instead of göründü for "appeared"?

While görünmek means "to be seen" or "to look" (e.g., "you look tired"), belirmek specifically means "to emerge," "to become distinct," or "to materialize." It perfectly paints the picture of mountains slowly coming into focus as the sun rises and cuts through the dark or fog.

Why doesn't the verb have a plural suffix (belirdiler) to match dağlar?

In Turkish, when a plural subject is non-human (like mountains, objects, or animals), the verb almost always stays singular. You typically only use the plural verb suffix (-lar/-ler) when the plural subject is human.