I am very new to Character Animator, so obviously still learning the fundmentals, but I had a professional create my puppet and background -- and whenever I go into "record" mode, the puppet goes out of frame (see attached screenshot) and I have no idea how to make it centered again. I'm sure it's a simple fix, but I can't figure it out. Any guidance would be deeply appreciated!
Yes, its a simple fix (and common problem). What you need to do is adjust the "Transform" behavior to set the "Scale" value to something smaller than 100%. I often find 60% works pretty well.
These days there is also a camera track - if you are going to use the camera track (e.g. to zoom in and out during a scene) you can do it that way as well. (But I am old school - Camera did not exist when I started, so I use Transform-Scale still, lol!)