What you call weeds is a nice-to-care-for ground cover. It probably would need a good hands-and-knees weeding twice a year if there is good rainfall. Also, it can handle some shade, but I don't know if it can handle the shade under those trees.
The trees are great. Building a wooden bench, that makes an octagon around each, is an idea. Potted Coleus or Impatiens could even go on some of the sitting area.
Putting in a hedge of fruiting bushes or dwarf fruit trees always adds more value to a home than a lot of lawn does and takes up some room along the edges, or divides a large yard into two or more "rooms." Lawns are expensive and mostly practical as playing areas. For a walk, people like tree-lined areas, or beaches better than lawns. For picnics or just sitting int he breezes, the benches, and a flower or vegetable area, are more fun.
If the soil is somewhat toxic with sprays, add some zeolyte products. See DirtDoctor.com.
Whatever you decide, do not put in a bunch of permanent masonry structures in. A small patio with a nice outdoor grill could work, but first figure out how the sun moves during the course of the year. You local Extension Agent, look in the Yellow Pages of the phone book, can help. Nothing takes a much time to develop as a nut tree, so if you have any interests in that, consider that placement first, and consider pollination issues.
Really, grass is a non-productive are unless you need a volleyball area or have chickens. Even goats need a lot more than grass or hay.