I'm preparing to start my CFI training and have almost exclusively logged all my time from the left seat of a Cessna 172 or 182. Is there a standard training program to transition to flying from the right seat? If not, how do people generally prepare, in an effort to make this less opinion-based, what are some common errors that pilots might make when learning to fly from the right seat?
First, congratulations! Earning a CFI certificate is a real undertaking.
For a literal textbook answer to your question I suggest opening this PTS pdf (assuming FAA CFI) and searching for "common errors" until you find yourself in relevant (to you) sections. There are many examples.
On a personal note, I found that talking through every action made maneuvers go smoothly and let the instructor/examiner know that what I was doing is intentional. It also forced me to follow my own directions so, since I knew what was supposed to happen, I just said it aloud and then did it. It sounds silly but it worked well.
The transition to right seat flying is simple. You'll have it down in an hour or two. Nothing to it. It's actually kind of anticlimactic. Which plane are you using?