Learn Finnish: Sinun kannattaa viilentää keltaista sävyä.

It is derived from the adjective viileä (cool). By taking the root word and adding the transitivizing suffix -ntää, it becomes an active verb meaning to make something cool. This suffix tells you that a subject is actively changing the state of an object.

If viilentää means that someone is cooling something, how would I say that the shade cools down on its own?

You would use the intransitive equivalent viilentyä (to get cool). The -ntää suffix always implies an active doer (transitive), while the -ntyä suffix means the change happens by itself (intransitive).

Why is the word for you (sinun) in the genitive case instead of the nominative sinä?

This sentence uses the verb kannattaa, which acts as a necessity verb here (meaning it is worthwhile or you should). In Finnish necessity structures, the person who needs to do the action is always put into the genitive case.

Why is the object keltaista sävyä in the partitive case?

Adjusting or changing a color shade is treated as an unbounded or ongoing process. Because you are tweaking the shade rather than completing a universally measurable, final action, the object takes the partitive case.