I love the Grand Canyon. Have hiked most of the trails many times never get tired of going there. However with that being said a week there is quite a while unless you are going to do some backpacking.
Day one - do the pull offs starting or ending at desert view, even stop at the museum/ruins west of desert view.
Day two- get on the shuttle and head to hermits rest hike back to the village about 7 miles. It for the most part runs along the rim. You can always bail on it and catch the shuttle back to the village at one of the numerous stops.
Day three -hike down into the canyon via bright angel (down to Indian gardens campground or perhaps even plateau point 4.5 & 6 miles each way but obviously you can go for as long or short as you like. Personally I prefer the south kaibab trail go down to skeleton point or tipping point or again as long or as short as you like. Bright angel more crowded but more services ie water south kaibab a bit steeper less water access but you have better views of the canyon for sure. Bright angel you hike down a side canyon and can barely see the main canyon until you get several miles down. South kaibab you can see the main canyon shortly after starting. June can be hot but start early or very late take plenty of water and rest when needed.
You can check out the village in the evenings, make sure and take a good look at Hopi house in the village yes it is a gift shop but really look at the architecture inside and out, same with desert tower and the kolb studio. Beyond that it is just more hiking or biking(on the rim)
North rim really to far for a day trip but is possible however over 200 miles each way
I would blast up to Bryce a few days there or the general area. Bryce has some good hiking, is higher in elevation (ie cooler) can be mostly seen in a single day or dang near every bit of it in two
Very scenic if you do make sure and go to one of the view points for some stargazing at night