To weed a garden requires knowing what to weed out. A weed is defined as "a plant out of place," which means that most any plant could be called a weed if it's not growing where you want it to.
In this garden, the preference is for plants that are native to the region, have some attractive attributes, don't grow into trees that will shade everything else, and don't spread aggressively by seed or rhizome.
So let's look at this photo of plants that popped up this spring. It's a mix of native and non-native species--Virginia creeper, willow herb, wood sorrel, nutsedge, violet, and one seedling of cutleaf coneflower. Since they're in the middle of a garden path, they all came out. Virginia creeper (five leaflets, lower lefthand corner) is a native vine that's fine for untended areas, but much too expansionist for a garden. Nutsedge (grassy looking leaves, light green) is a non-native sedge that pulls up easily but keeps popping up, requiring eternal vigilance. Wood sorrel (clover-like leaves) is a ubiquitous presence in gardens and greenhouses, with a little yellow flower and acid taste. Willow herb (narrow leaves in pairs) is a weedy native that sprouts abundantly from seed. It has a promising form but miniscule flowers. Violets are attractive and tasty, but not in a garden path.
Only one plant was worth potting up for later use--the cutleaf coneflower seedling, its two broad, oddly lobed leaves visible here in a blowup from the original photo. If given a good place to grow, it will become a tall, stately wildflower bedecked with bright yellow flowers. The seedheads in turn attract a second wave of yellow, in the form of goldfinches vying for a snack.