I just go with a simple, budget-friendly plan to start. Mow at least once a week on the highest mower setting. This will discourage weed growth and reduce water requirements. Mulch mow the clippings back into the lawn. This reduces fertilizer needs and water requirements. Fertilize now with any standard lawn fertilizer - organic will work fine. Don't bother watering now, but once summer hits, water once a week with 1" of water. Any easy way to measure one inch is to take an empty tuna can and put it out when you start the sprinkler. When it is full, you are done. If you really do have a ton of weeds, you might consider a 1 time treatment with weed-b-gon. The hose-end sprayer kinds covers a large area quickly. It definitely isn't organic, but it is cheap and fast and won't interfere with your ability to chose an all-organic program down the line. If you do those things, your lawn will improve dramatically from now until the end of summer. It definitely won't be perfect, but it should be a ton better. At that point, you can decide if you are happy with "improved" or want to start over with premium seed and go for a showcase quality lawn.

You could try seeding with a wildflower mix mixed with sand and just let it go. I use the false strawberry as a living mulch in part of my yard then pull as I put in a plant. I have a problem with my front slope to the road 10 ft tall. I am slowly filling it with plants that need little care but expand by their roots such as peonies, various goldenrods, asters etc. The only problem I am finding is the trees that sprout on the slope. I have slid down the slope several times while cutting them out.

Possibly Epipactis helleborine, Broad-Leaved Helleborine. I can't tell you how to get rid of it. It is a threatened native in the UK and I'd love to have it in my garden. Unfortunately it is an invasive species in the US.

They look like hackberry seedlings to me. The blog post I've linked has an excellent clickable picture of a hackberry seedling, it's the 9th picture down. Here is a link that might be useful: The Birth of Trees