How early do you have to get to Yellowstone to get a spot in the first come first serve campsites?

I'm in a group of 5 traveling to yellowstone from Michigan next month. I wanted to do one of the first-come-first-serve sites for camping, but I don't want to drive thousands of miles and not get a spot. We are leaving on a Sunday early morning (~5:00 AM) near the end of July and driving non-stop until we get there.(23 hours, according to google maps) and staying for a week. What campgrounds should we target first and is Monday a good day to arrive, or should we leave saturday and try to get there on sunday?