Learn Finnish: Kotka saalistaa savannilla. - The eagle is hunting on the savanna.

You would use the verb olla and the fifth infinitive form of saalistaa, which is saalistamaisillaan. The full sentence becomes Kotka on saalistamaisillaan savannilla. The fifth infinitive suffix -maisilla- is combined with the third-person possessive suffix -an to match the eagle.

The verb saalistaa looks like it's related to a noun. What is the root?

It comes from the noun saalis, which means "prey" or "catch". Adding the verbalizing suffix -taa is a common way to turn a noun into an active verb in Finnish, so saalistaa literally means "to gather prey" or "to hunt".

Why do we use the adessive case (savannilla) instead of the inessive (savannissa)?

In Finnish, open outdoor areas and flat plains like a savanna, field (pelto), or yard (piha) typically take the outer locative cases like the adessive (-lla, meaning "on") rather than the inner locative (-ssa, meaning "in").

If we added an object to this sentence, like a mouse, what case would it be in?

The verb saalistaa generally takes a partitive object because hunting is viewed as an ongoing, continuous process. For example, "The eagle is hunting a mouse" would be Kotka saalistaa hiirtä, where hiiri is put into the partitive case.