Day one. Drive to Salida CO via US 285 out of Denver. Only about three to four hours, but a good shake down route without high passes for that first day out. Salida has a Walmart and other stores if you need more supplies. (You can expand your camping search area in Buena Vista if necessary). Lots to do around Salida, but exploring the old down town by the river is a hit with adults and kids alike.
Day two. Drive to Montrose/Ouray area via US 50. Camp near these towns or head up to Black Canyon of the Gunnison NP for the night. The trip takes you past Cimarron, where you should take the 2-3 mile drive into the canyon to the base of Morrow Point Dam.
Day three. Drive US 550 through Silverton to Durango. With a TC, you shouldn't have any problem with this section. Camp around Durango. Make sure you allow yourself several hours to explore Silverton. A real old Colorado mining town with wooden sidewalks and dirt streets. This is one of the most beautiful segments of mountain highway in CO. Plan on lots of picture stops.
Day four. Spend part of the day exploring Mesa Verde NP. Depending on the age of your kids, they might get a real kick out of this, plus a little education slipped in there too! Try camping around Cortez (McPhee Reservoir has NFS CG's nearby).
Day five. Drive along the South Rim of the Grand Canyon and then camp south of the park, as the CG's in the park are probably filled (check on your specific date, you might luck out). You might check around Flagstaff also if you need to expand your search area. This will be a longish day, but you can see a number of the viewpoints along the canyon rim.
Days 6 and 7. You have the main options of I-8, I-10 and I-40. Where you want to visit will dictate which of these roads to take and where you'll want to camp. Depending on how late in June you are talking about will probably dictate where to go based on how much heat you want to deal with.