You're in for a really boring drive.. Honestly you'll have a far better time just spending the money in Vegas that you would've wasted on a rental car .
I'll do you a favor and just describe your drive for you, I did a similar drive not too long ago:
Leave LA, wait in traffic until you get into the cities and suburbs known as the inland empire, miserable.
Pass through the inland empire into the Southern California desert, you'll see some wind turbines but thats about it.
Pass into Arizona, a state that is hot as shit and almost completely covered in desert. You'll see some cool cactii, thats about it. The Grand Canyon is worth seeing if you go to Vegas and make a day trip out of it, but not worth driving across the country to see. Also, if you're driving in August, it will be very hot.
Pass into New Mexico, more desert, almost the same as Arizona.
Pass into Texas, more vegetation than New Mexico or Arizona but its still desert.
Reach Oklahoma, begin to see nothing but rows upon rows of corn and other agriculture.
Get to Missouri, I've never been there, but I'm willing to bet it isn't too different from Oklahoma.
Enter Illinois, see all the redneck areas of the state before finally arriving in Chicago, which is quite nice from what I hear. So after over 30 hours of driving between LA and Chicago, you've managed to tour through some of the most boring places in the US.
I'd advise you against your idea. Airfare is very cheap at the moment, just fly from LA or Vegas to Chicago, you'll thank me for it. Or... if you really want to go through with this whole road trip idea, drive up along highway 1 from LA to San Francisco. It runs up all the way along the California coast, so you'll have plenty of nice beaches with pretty girls to stop at along the way. You won't want to leave some of the places you stop at.