It is traditional, when returning from a trip, to bring small presents for your coworkers or friends. These are typically small food items, like chocolate or candy or crackers, of which you can find boxes at any train station. If you go to Tokyo, that's probably far enough that you should bring omiyage for your coworkers. But if you only go to Aomori or Yamagata, maybe you can skip it.
Some ALTs teach at many schools, and it would be unreasonable to bring omiyage for several schools, so use your best judgment.
ALTs first coming to Japan might bring a present for their new bosses, but it's not necessary. When you're moving to a new country, everyone understands that you have a lot to think about, and they don't expect presents from you.
Insider's Tip: If you don't tell anyone that you went on a trip, no one will be expecting omiyage. This approach might be more difficult if you are gone for an extended period of time.