OhTeeDee, Yes, Judaism believes in such things. However, as I mentioned, it’s not absolutely required to believe in them, because there were some rabbis (mainly before the Zohar) who wrote against the notion of reincarnation, and also the Rashash, who lived less than 300 years ago.
The reason you didn’t hear about it in your MO education is because MO Judaism in general (like most of the Litvish world) stays away from kabbalah, and kabbalah is where the teachings about reincarnation come from.
However, that’s not to say there aren’t major modern or Yeshivish rabbinical authorities who have openly discussed it. For example, the Vilna Gaon wrote about gilgulim in his interpretation of the book of Yonah.
All or most chassidim are probably familiar with the concept of gilgulim, because unlike other groups, the study of kabbalah (through chassidus) is normal and widespread. However, even among them, the topic doesn’t come up all the time.
In my view, the main practical consequence of believing in gilgulim should be to provide further motivation to perfect yourself in this life (eliminating negative character traits and fostering positive ones, as well as following mitzvos and studying Torah), because if you don’t fulfill the mission of this gilgul, you may have to live again (and your life might not be that pleasant next time — you never know.)
Another thing is that there is a teaching I’ve heard from various sources (I don’t know the original source), that if you die without Hashem having forgiven you for a sin between man and God, you will have to spend some time in gehinom, but if you die without having a person forgive you for a sin you committed against them, then you will have to have another gilgul to atone for that.