Answer: New lawn grass can be damaged by broadleaf weed killer, so you will need to wait until fall for that. If you used a starter fertilizer when you planted the seed then it is still too early to fertilize again. It is just as bad to overfertilize as it is to underfertilize. Check the label of the product you used for its expected duration.
If you have not fertilized yet, I would suggest you run some basic soil tests to check fertility and also the pH of the soil. Use the test results to determine how much to fertilize and if you also need to add lime to adjust the pH.
Your lawn should be mowed often enough that you do not remove more than one third its height when you mow. This can mean mowing more than once a week when it is growing quickly. Cutting off over half the height at once will stress the grass. Set your mower as high as possible and mow it now; then set it at three inches for the rest of the summer.
Your local Cornell county extension should be able to help you with the soil testing and interpreting the results and developing a lawn maintenance plan. In the meantime, you may find the following information helpful. (You may need to cut and paste the complete url to make it work correctly.)
Good luck with your lawn!
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