Don't worry too much about the labels, but you need to know what they are in order to converse with others.
Action verbs are the easiest, although you will encounter some disagreement about what "action" really means.
Non-action verbs can be divided a number of ways.
Some divide them as state, sense, desire, possession, or opinion. Others use other classifications.
One of the most important classification systems for non-action verbs is whether or not they are linking verbs.
Linking verbs are different because they take predicate adjectives or predicate nominatives instead of direct objects.
In most cases, being verbs are linking verbs, but some sense verbs and other "mental" verbs can also be linking verbs.