Welcome to the site! I do indeed have a couple of ideas which you may like to consider.
This is not the most direct route, but the extra mileage is, I reckon, well worth it.
Take AZ-64 out of the park (I am presuming you're visiting the South Rim) until you hit US-89 where you hang a left and head north towards Page AZ. There are numerous Navajo stools all along this route selling goodies which are well worth stopping at. There is an amazing store along this route also, I sadly cannot recall it's exact location at the moment, but you'll know it when you see it. It's the one with the huge car park full of Trek America and Suntrek buses!
Anyway, before you reach Page, turn left onto US-89 ALT for several miles until you come across the Navajo Bridge. Check out the bridge (there is a small visitor center IIRC but the thing to see is definately the bridge - walk across it and look down!) then continue a short distance before turning right towards Lees Ferry. This road dips down until you are level with the Colorado river and you can paddle in it if you feel inclined!
Afterwards, retrace your steps, across the Navajo Bridge, along US-89 ALT and, when you reach US-89 again, turn left towards Page once more. Before you reach Page, look for a poorly marked turning (on your left) for the Horseshoe Bend. It is a short hike across sand but is well worth the effort - especially at sunrise or sunset. I know this is far from the first time that I've recommended this, and the regulars will probably roll their eyes at a further mention, but it is that good!
If you end up spending the night in Page, be sure to eat at 'Kens Old West' - a great experience!
Next stop is the Glen Canyon National Recreation Area where you must visit the Glen Canyon Dam. I found this far more impressive than the Hoover Dam - probably because you can view the dam from a bridge across the canyon - this gives you a real impression of the size of the thing.
As you leave the area and head towards Durango be absolutely sure to visit Monument Valley and take a tour with one of the Navajo guides. I didn't do the tour myself but I am told I really missed out.
Finally, looking at my map, it would appear Mesa Verde National Park is pretty much on your route and may well be worth a visit if you have time.
Hope this is of some help to you. Be sure to come back afterwards and let us know how this, as well as the rest of the trip, worked out for you.