To remove the hard plastic car emblems themselves. Most people find using dental floss or fishing line effective. I have a very steady hand so I actually speed this step up by using the sharp end of a flat screw driver to catch the edge of the plastic emblem and lift up without touching the paint work. If you want to use this method and lack a steady hand, you can lay some blue painters tape down to protect the paint first.
As long as it doesn't have locating pins..
There will always be some form of double sided tape underneath the hard plastic car emblem. This stage of the car emblem removal process will be the most difficult and most damaging. There is no secret or simple way to remove the double sided tape and adhesive layer. I usually pick off, and rub off the spongy double sided tape to reveal underlying adhesive residue. Removing the adhesive residue will require much rubbing with a rag dampened with some form of
solvent that will break down the adhesive.
Having accepted the fact that during the car emblem removal process, you are doing some damage to your paint and you will need to polish the area afterward. The lighter the color of car you are working on, the less critical this step will be.
Dark colored paint shows every blemish and flaw; light colors are much more forgiving. Unless you know how to color sand and use a buffer, you will be limited to how "perfect" the finished product will be.
Then take to detail shop to professionally polished...