Since nobodies thrown a good idea up for ya yet I'll tell my way.
I've CAREFULLY used a box cutter to slice the tape loose on emblems. Then a blow dryer to heat the remaining foam/tape and peel with my fingers till they feel like I've twisted the skin cockeyed! Use goo gone to clean the residue.
Factory Side Mouldings, I wouldn't know the best way. I've never tried to remove one while not messin up what's below. Always done as part of a repaint. I've never had a good time removing them either! They're on there real good! Usually end up screwin em up gettin em off cause they got an aluminum backing that gets bent real easy. I always end up gouging the paint with the razor or nicking the aluminum backing somwhere, etc.
Have Fun!