Answer: Surprisingly, the best time of year to do this is late summer to early fall with very early spring being the next best time. It is already a bit late in the season for extensive lawn renovation, but to redo the area you would need to remove the existing weeds by using an herbicide containing glyphosate (read and carefully follow all of the label directions. You would run some basic soil tests to check fertility and pH and then add fertilizer and/or lime as indicated by the test results. Then loosen the soil down about six inches and work in some organic matter such as compost, then level the area and rake it smooth. At that point you could lay sod or seed. The area must be kept evenly moist while the new grass establishes.
Your local county extension should be able to help you with the soil tests and interpreting the results, suggesting the best grass varieties for your yard, and developing an ongoing lawn care program.
With such a small area, I should also mention that it may be more efficient and lower maintenance to use a ground cover rather than lawn grass.
Enjoy your new home!
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