Learn Russian: Пользователи читают новости в разных социальных сетях. - Users read news on various social networks.

Сетях comes from the noun сеть (network). Because it ends in a soft sign (-ь), it is a soft noun. In the prepositional plural, soft nouns take -ях instead of the hard ending -ах.

Why do the adjectives разных and социальных end in -ых?

They are describing сетях in the prepositional plural. For most adjectives, the prepositional plural ending is -ых. You only use -их if the adjective stem ends in a soft consonant or a specific letter governed by spelling rules (like к, г, or х).

Why does the sentence use в (in) when the English translation says "on" social networks?

In Russian, digital spaces like the internet (в интернете) and social networks (в социальных сетях) are treated as environments you are "inside" of, so they take в (in) rather than на (on).

The word for "news" is the direct object here. Why does новости look like it is in the nominative plural?

Новости is indeed in the accusative case here because it is the object being read. However, for inanimate nouns like новость, the accusative plural form is exactly the same as the nominative plural form.