Answer: First of all I would suggest you run some basic soil tests to check the fertility and soil pH. The test results will tell you whether you need to fertilize and/or add lime. There is no way to judge this without testing and it is critical to the success of your lawn.
Generally speaking it is a good idea to add several inches of compost worked into the soil. It is smart to similarly test the compost so you know what type of material you are adding, it can vary. Spent mushroom compost can sometimes be used, as can old rotted leaves from a municipal leaf dump, or well rotted stable manure/bedding and so on. Your local county extension should be able to help you with the soil tests and be familiar with the local compost products.
I would caution you about disturbing soil in the root zone of the oak as extensive changes to grade and/or root damage could stress or kill it. Also, lawn grasses grow best in full sun all day or for at least a bare minimum of half the day. If the area is too shady, you may want to consider planting a shade tolerant groundcover instead.
Finally, the best time of year to do lawn work is late summer or early fall. The reason for this is that the soil is already warm enough to germinate the seed, and the fall season is usually naturally rainy so the grass has the best weather to grow and become rooted -- the fall season and again in the spring -- before being stressed by the heat and dry spells of summer. So if your seeding now does not work as well as you would like, you have another chance this fall.
In the meantime you may want to dig out by the roots or spot treat perennial weeds with an herbicide containing glyphosate. Read and follow the label directions.
Lastly, please understand that a lawn takes time and effort as well as ongoing maintenance. The true "golf course" look at golf couorses is achieved by highly trained turf care professionals using specialized grass varieties and following the latest research in specialty turf management. A home lawn will probably never look quite the same.
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