Answer: The best time of year to redo a lawn is late summer to early fall.
In the meantime, mowing correctly can help the grass crowd out many weeds. Set the mower blade at three inches and mow often enough that you never take off more than one third the grass height per mowing. In spring, this can mean mowing more than once a week. When you have dandelions blooming, bag your clippings and put them in the trash so you are not returning dandelion seeds to the lawn area.
Also, run some basic soil tests to check fertility and pH. This will tell you how much fertilizer to use and if you need to also add lime. (Dandelions tend to grow in soil that needs lime to raise the pH.)
Your local county extension can help you with the soil testing and interpreting the results. They should also be able to advise you on redoing the lawn, whether by overseeding or by renovation later this season. In the meantime, you may find some helpful information about lawn maintenance, weed control, and starting on their web site. You may need to cut and paste the complete url into your browser to make it work correctly.
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