Learn Italian: Abbiamo viaggiato per un mese.

We use per because the action is completely finished ("for a month"). You can tell it's a completed duration because the verb is in the past tense (abbiamo viaggiato). If you were still traveling, you would use the present tense with da: Viaggiamo da un mese (We have been traveling for a month).

Since viaggiare (to travel) is a verb of movement, shouldn't it use essere as its helper verb?

You are right to remember that many verbs of movement (like andare, partire, or venire) use essere to form the past tense! However, viaggiare is a notable exception. It always uses avere. So we say abbiamo viaggiato, never siamo viaggiati.

Why doesn't viaggiato end in an "i" since the subject "we" is plural?

Because the helper verb here is avere (abbiamo), the past participle (viaggiato) doesn't change to match the subject. It just stays in its default -o form. You only change the ending to match the subject when the helper verb is essere (for example, siamo partiti = we left).

Could I say durante un mese instead?