I want to be able to have the camera rotate around an axis, while still remaining pointed at the center of said axis. The camera should pan around the model in a circular motion with the model always in the center of the view. How do I do this?
You can parent the camera to an Empty to do this. While in the 3D Viewport, press Shift + S and select Cursor to Selected or Cursor to Center. Add an empty (Shift + A > Empty > Plain Axes). Parent your camera to that empty by selecting the camera first then hold shift while selecting empty. Press Ctrl + P and select Object (Keep Transform), now you can rotate the empty 360 degrees based on Z axis so your camera will pan around in circular motion.
Hope you find this useful!
Another trick that might help is to track the camera to a target.
If you have a specific object you want to orbit around and the motion doesn't have to be absolutely circular then you can add a Track To constraint to the camera with the object in question as your target. If you keep Space as World Space <-> World Space, you'll want to switch the Up: to Y and the To: to -Z.
Instead of directly tracking to the object in question you could add an empty of course, and make sure the empty is where it should be. You then can move the camera freely around, it will keep tracking its target. Set your key frames for camera location where you need them, rotation is handled by the Track To constraint, keeping the camera trained on its target.
The quick way to do this is to first
select your target, then Shift +
select your camera. Shift + Ctrl + C will give you a constraint popup, select Track To. The first selected target will be automatically assigned the target for your constraint.