a transitive verb can have a direct object linked to it in a sentence
ex. In the sentence 'he lifted his pencil', lifted would be a transitive verb
an intransitive verb can't have a direct object linked to it in a sentence
ex. In the sentence 'he looked at his paper', looked would be a intransitive verb because you need the at for the sentence to make sense
an action verb shows physical or mental action, telling us what the subject is doing
ex. In the sentence 'he became nervous', became is an action verb because it is describing how his mental state changed
a linking verb can connect the subject with information about the subject. they do not show action, instead showing a connection or relationship.
ex. In the sentence 'It was a hard problem', the word was connected the subject, it, with more information about it, that it was a hard problem.