Answer: You actually have three options: you can simply overseed the lawn after removing the weeds and moss (the easiest solution), or you can revamp the entire area and then either lay sod or seed. It will be necessary to rototill and remove existing vegetation in preparation for sod installation or seeding. So, if you're feeling ambitious, rent a tiller, rake out the debris, level the area and have sod installed; or broadcast seeds.
Once you've prepared the area, sod will give you the quickest results, with the bonus of a weed free lawn for quite some time. It's also the most expensive option. Starting with fresh seed, a seeded lawn will green up in 7-10 days and, as long as you keep weeds down, the lawn will grow lush and thick by the end of the summer.
Best wishes with your new lawn!
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