Hey Sarah :)
Weeds are good for you, for your soil, and for your intended crop, I reckon! Read below ...
As far as I know, weeds can't be stopped from growing, if the seeds are in the soil (unless the soil is heated/baked to destroy all the seeds - but heating/baking to this high a temp is not good because it also destroys the benefical microorganisms in the soil).
As far as I know, the best - but slower and more labour intensive - way is to remove the weeds as they come up, pull them out (roots & all). Or at least get out as much of the plant as you can (even if some root remains, it'll be a few weeks until that plant appears above ground again).
Having a multi-culture - rather than a mono-culture - of plants is generally better for each plant and for the soil - eg companion planting - but there are some plants that don't do well if together - you can find info about this if you research. So, having these "weeds" in your veggie garden patch can be (and is) a good thing for your intended crop and for your soil. Some of these "weeds" have very deep roots which bring up nutrients (minerals) from deep down the soil, so if you use these "weeds" as mulch, your soil will improve. Also the more plants there are (the more plants of various kinds, including "weeds"), the more (in different ways) your soil will be "conditioned" / enlivened by the plant roots permeating through the soil - which is good for the intended crop -- much like how it is in a natural forest - research forest gardening.
I have a similar problem in my small suburban garden, and I (like you) make use of the "weeds" - grass, nettle, dock, dandelion, puha, catsear, hawksbeard, cleavers (bidi-bid), chickweed - I juice / blend / eat (in salads) these "weeds" - which are more nutritious and more sustainable etc etc than the modern commercial/hybrid plants that most ppl intend to grow! True! :)