The pod system is to try to keep the teams as close to home as possible the first two games and to limit travel times. The pods have no bearing on which region you are in, other than the teams that you play in your pod are in the same region that you were assigned to. There will also be teams in your pod city, that you do not play, that are from another region. Obviously some teams have to travel, and it is usually done in some sort of order of the regional seedings with deference towards the higher seeded teams (top 16).
So let's say, for example, Syracuse ends up in the West region, and we are the #2 seed. That year there are first round games in Albany, NY. We would play our first two games in Albany, NY. We would have the teams that line up with our seeding from the West region play in Albany. So our first game would be against the #15 seed in the west region. Now let's say the #15 seed in the west is Utah. Then Utah is obviously going to have to travel. They are just lucky to be in the tournament. Syracuse, because they are one of the top 16 teams, would get to stay near home. Now also in Albany would be four teams from another region, let's say the East region. Those games and the West games are not at all related other than they are in the same arena in the same city. After the Albany weekend, the teams then travel to their respective regions for the next two games. So at that point, Syracuse would have to travel out West to where those regional semi final and final games are being held. The east teams would travel to Syracuse if that is where the Eastern Regional games are to be played.
It is very complicated. I hope this helps you to understand the basic concept. Even what I have laid out is not how it always happens.