If you are a person who doesn't mind introducing poisons into the environment, you can go the roundup way. However you have to realize you would need to spray the round up for an area at least a foot beyond where you see the weed grass, because grasses spread by runners which will be in the soil past where you see them. EVERYTHING in the sprayed area will die. Then you would need to re-seed or resod the dead area.
If you are going to do this, look for glyphosate, which is the active ingredient in RoundUP, rather than brand name Roundup. Apparently some of the things in RoundUp that are listed as "inert" ingredients are environmentally worse than the glyphosate.
If you don't want to lay down poisons (if you type RoundUp into search box in upper left of most pages you will find information about it and pros and cons), you could try putting down corn gluten meal. You would want to do it NOW and again in early spring. It is a pre-emergence herbicide that does not kill any existing weeds, just prevents new seed from germinating. It also acts as a nitrogen feed to the soil and is not harmful to the environment.
You could just hoe out all the offending stuff and reseed. You are right putting dirt over it probably wouldn't help. If there are patches you want to get rid of, you can smother them with a thick layer of newspaper, then put dirt on top of that and seed into it. (do a search on sheet mulching to read more re this technique) But just as with the RoundUp, you would have to smother a bigger area than where you see the weed grass or it will come back from runners.
On the other hand if it is "nice and green and soft to walk on" you could just live with it. If it is bent grass, people make nice looking lawns of that stuff. Since I don't use any chemicals, my "lawn" is a mixture of weeds and grass. I keep it all mowed pretty short and trimmed and it looks just as nice and green and even as anyone elses...