Ho passato la maggior parte del tempo ad imparare il vocabolario e a studiare la grammatica
Ho guardato ovunque sperando di trovare le chiavi che avevo perso
Il cane corse freneticamente, abbaiando forte come non mai
in the first sentence there is no use for the italian 'gerundio', because the actions of learning and studying aren't following any other previous action.
you can see the difference in the other two: I was looking while hoping, the dog was running while barking.