Lots of moss is mainly down to poor drainage, use a moss killer and rake the dead moss out, fork the area where the moss was and sprinkle grit down the holes and then reseed the area, you can seed again once you have some growth(its called overseeding and is used by groundsmen), as for the other weeds use a weed killer you can directly spray on to the plant, once its dry it will be harmless to pets and children.
Dave.