Linking Verb=A verb that connects with the subject to a word that describes about the subject. For example, "They are a problem." "They' is the subject, "are" is the linking verb, and "problem" is describing "They."
An Action Verb=A verb that expresses action. An example would be "The boy ran." "Ran" would be the action verb.
Combining these together, a sentence could be, "He is fast, and ran like the wind." Or, "You are smart, and you ace all your exams."
In the first sentence, the linking verb is "is," and the action verb is "ran." In the second sentence, the linking verb is "are," and the action verb is "aced."
I hope this helped. :)