$500 budget won't get it done. But, having said that, here's my opinion:
Don't worry about the weeds until you have something to replace them with. Anything green is better than nothing. The single most effective piece of equipment for low budget turf is the First Products Aeravator. I would aeravate and overseed (slow, less than 2 MPH) the entire field with Tryplex Perenial rye at 4 - 6 lbs. per thousand square feet. A split rate in 2 directions is best. the benefit to the aeravator over the slit seeder is that it relieves compaction. Even if you slice seed I would aeravate to relieve compaction. If your aerator won't go in the ground at least 2" I wouldn't depend on it as preparation for slice seeding. If you have the budget for topsoil aeravate the areas before adding the soil and aeravate and overseed after adding the soil. bare areas and topsoiled and seeded areas will have to ber mulched with hay or other acceptable mulch product or the grass wn't grow.
Now, I am probably getting ahead of myself since a soil test is important prior to seeding to determine the fertility of the soil. soil pH is most important. At the very least you want the soil over 6 and less than 6.8. Additionally the potassium and phosphorous levels should be addressed based on soil test recommendations along with a light application of nitrogen at the time of seeding. Good luck!