Much depends on where at the Canyon you want to go to. The main visitor sites are at the North and South rims, about 15 miles apart as the crow flies. However, there is no way to drive directly between them (it's about a 200 mile drive).
To get to the North rim, you have to go through Utah. North Rim is less "touristy" and the weather is cooler (the altitude is about 9000 ft.). Make reservations well in advance, as the lodge is about the only place within a hundred miles to stay (or eat).
South Rim is where most people think of when they say "Grand Canyon". It's easily accessible (only an hour north of Flagstaff and Interstate 40). It's only at about 8000ft, so is slightly warmer. I'd suggest may or june for both north and south rims.
You can hike to the bottom of the canyon from either rim. I've made it to the bottom (Phantom Ranch at the River) and back in one day (it was dark by the time I got back to the rim) but it was one hell of a hike/run. Do not attempt it unless you are in great shape and used to mountain hiking. The climb out is grueling once you are tired.
Do you have a 4x4? If so, there are tons of scenic dirt roads and trails in the area.
Sunset crater, north of Flagstaff, is well worth a visit. It has a volcanic cinder cone and lava flow (from 500 years ago) and also Indian ruins, plus a natural blowhole. (changes in barometric pressure cause strong air inflows or outrushes)
Definitely see Meteor Crater, too. Further east on I40, don't miss the Petrified Forest.
If you can, take a detour to Southern New Mexico to see Carlsbad Caverns. It is one of the most spectacular sights on the planet. The sheer size is incredible. If you go, stay near the park entrance, and go early on a weekday morning, right when they open. You can hike from the surface down to the main chamber. It's an easy hike, just 2 1/2 miles, though much of it steeply downhill. If you can go when there are no people around, the experience of being alone in those chambers is well worth it, and makes the hike an utterly unforgettable experience. Once you reach the bottom, another two-mile hike takes you around the perimeter of the main chambers (yep, it is that big). Once done, you can take the elevator (no kidding!) back to the surface. In fact, you have to, as the trail in is one-way. You can take the elevator both ways, but I recommend the hike. It is incredible.