Not to be a downer, but it seems that this trip is not planned well, especially involving the huge distances and the extreme temperatures of the southwest deserts in mid July.

Desert Dancer (note desert) is right when she said "you [New] English folks really do have a hard time getting a handle on the scale of the American west...".

Also, the area people usually think of when they say Grand Canyon is close to the center (east-west) of Arizona and is reached by going north from Highway 40 (old Route 66). The best you can do, within a reasonable drive from Las Vegas is perhaps Meadview, about 100 mile drive from the Vegas strip, and that is not the exiting part of the Grand Canyon. The Vegas-Meadview trip goes over the Hoover Dam, a great experience in itself, but that area is probably the second hottest part of the US, behind only Death Valley. Expect temps near 120 F.