<what's the correct way to remove interior paneling?  It seems a lot of
the panels are held in place with plastic clip thingies.  What's the
best way to remove them to avoid damaging the clips/panels>

With a "U" shaped tool made for removing car trim panels. Try to use a screwdriver and you'll likely break the panel. As for the clips, no matter how careful you are, some will break because they are brittle from age. When that happens, go to Lowes (NOT Home Depot) and you'll find suitable replacements (though not original) in the specialty hardware section.     Jeff