Kelly always read the label on anything you spray on your yard. Round up is a nonselective herbicide, it kills everything. Spring seeding cool season grasses is difficult to do in the spring and keep alive in the summer, not to mention the competition from summer annual weeds. Sod is your best option with the best chance of success. Tall fescue or zoysia sod. Tall fescue is probably your best option, but I would look into both. Tall fescue sod is around 40 cents a square foot if you install your self. If sod is not in the budget then look into Prime Time or Bullseye tall fescue it will cost around a hundred dollars for a fifty pound bag and 50 lbs will cover around 6000 square feet. When you decide what kind of turf and if your going to seed or sod post again and I'm sure you will get plenty of responses on a step by step process, but either way it's going to be a fair amount of work.

Frank -
Now that you have no lawn, don't reflexively plant more in exactly the same places.

Take some time and consider less lawn and more shrubs and flowers. And maybe a wider sidewalk.

Yup, read the label! Ugh. I read only what I wanted to read.

Thankfully I limited my stupidity to the very small back yard and only killed about a third of that.