What's an easy way to get my complements right?

One of the easiest way, in my opinion, is to learn how languages often follow a common scheme of cases.

Nominativ, Accusativ, Dativ and Genitiv are the most important and can be used as containers to easily memorise most complements.

e.g.: Dative gets its name from the latin "dare" which means "to give", so it usually refers to the recipient of something/some action (in english as well as in italian)

Here is an impromptu scheme of the main functions expressed by every case:

Nominativ -> subjects

Accusativ -> direct objects

Dative -> indirect objects

Genitiv -> belonging and specifications.