You'll be disappointed after reading this, so be prepared! Ultimately you have to learn each verb individually. You have to learn which case and which preposition(s) to use with which verb.

Take this example: the verb "помочь/помогать" (to help). It obviously requires direct object: when you help someone, you're doing the act of helping directly to him/her. So you would expect it to take the accusative case. Well, it doesn't. It takes the dative: помочь кому-нибудь.

That's not just one exception to the general rule. There are many more. The only way is to learn verbs one by one: their meaning, their conjugation, the cases and prepositions they take, etc, etc.