How can I remove these plants that seem to be at the root and not kill the rest of the stuff in the garden?
Just pull it slowly so that, you wont kill other plants..
If you are talking about the spreading groundcover with the colorful leaves you have two options. Fork it out whenever you see it or spray it with Roundup or Crossbow. A shovel just cuts it up and every piece will grow. It called Houttuynia cordata, 'Chameleon'.
I found this online. Sounds like you can get rid of it, but it will take commitment.
Here is a link that might be useful: Ask the expert