The best action verbs for elementary-age students are the ones they can physically act out and that are simple enough for them to understand and relate to. The most important point in teaching young children action verbs is simply to get them to understand that the word represents an activity.
Once young children can relate to the idea that words are used to describe actions, the students are prepared to use the verbs in their writing, which is an effective way to teach the functional grammar necessary for writing. By elementary school, children use action verbs in their speech every day, so the goal becomes helping them to connect the action and spoken word to the part of speech when it's written.
The topic of action verbs for elementary students lends itself well to classroom fun. A teacher might take an active approach to teaching verbs and have students act out a list of things they do every day, describing them as verbs. "Brush your teeth" could involve having the students pretend to brush their teeth, or "run" might have the whole class running in place for a moment. Another idea is to have students call out things they do every day, such as "walk the dog," and then have the class identify the verb in that activity.