75% of weeds in my backyard. best way to get rid of?

Not a green thumb at heart but slowly learning. We are stationed in a townhouse and 75% of the yard is infested in weeds. When I cut the yard it looks ok, but if I take a rake to it, all the weeds pull up and it looks terrible. Tonight we spent a couple of hours pulling weeds (lots of soil had to come up along with them), but we are afraid this will not end our problems. Housing will not do anything for us.

So..

What is the best way to get rid of all these weeds? Do we really have to go by hand through the whole yard pulling weeds, then lay soil where needed and plant new grass? If you need any other information, please let me know. Not sure what types of weeds they are. Most of them come out fairly easy. Some of the weeds have thick stems with white sap in the center (not that many in the yard) - I believe those are poisonous. There are also vines in the neighbors yard that creeps over. It doesn't help that they do not take care of their yard either (on either side of us).

Round up will kill anything it touches.

Start with weed'n feed, from any number of suppliers (Scott's). These are just mostly high nitrogen mixes with some herbicides.

Mind you, this is a quick fix. You could get more thorough and invest a little more time and energy into thatching and re-seeding.

Use a broadcast spreader, not a drop-spreader, or you'll leave linear marks in the yard where you didn't overlap.

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Sugarland Landscape Design