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Identifying and controlling weeds is process of elimination. Looking at a given part of the garden one must decide which plants to focus on to control first.
Rule number 1: Grasses should almost always be the first to go. Particularly the larger running grass like Quack grass. This grass has a larger leaf and has a distinct running habit (as opposed to clumping). Because these plants will run throughout the garden and into dayliy clumps, they should be addressed first.
Other running plants that should be removed : Nutsedge, Mugwort, Bindweed and Vetch (these are all running plants)
Rule number 2: Remove weeds that will get large: These include plants like Velvetleaf, Wild lettuce and biennial Evening Primrose. These will get very large and dominate your garden in that space
Rule number 3; Smaller plants might be left: Hawkweed, Violets and Wild Geraniums have nice flowers and don't grow so big as to crowd out daylilies. Though if growing too close to a daylily removal might be advised as it will compete some for water and nutrients.
White clover is good for the soil but will run, so keep that in mind.
Rule number 4: Some plants are persistent, Dandelions are a good example,  repeated cuts to the crown are needed to permanently eliminate them, so it might be worth weeding once or twice and accepting some dandelions in the garden.
Lastly Nature Abhors a Vacuum, better to have something relatively harmless growing than nothing as something will grow there!
So how would I manage the plot below?
First all running grasses would be string trimmed or weeded out. Other grasses next. At the same time all Dandelion shoots would be string trimmed or weeded at the same time.
The rest would be left unless they were growing too close to a daylily.
Next go around with a string trimmer or weeding would focus on any grass regrowth and patches of plants that are encroaching on the daylilies