You are looking at going into one of the world wonders of nature. So many NPs and other attractions, three weeks will start to sound like 'too little time'.

My first suggestion is that you check out the area on good paper maps. If these are not available locally, I would strongly urge you to purchase a Rand McNally road atlas - available from the RTA store at the bottom of this page. Order it now and you will have it in a couple of weeks. Once you have the maps of that area, you will begin to realise just how much there is and how often you can stop and explore. Bonus is, the most scenic routes are all hilighted.

... then head towards Vegas/grand canyon ...
Be aware that you will see many tours to the Grand Canyon advertised in LV. Few if any of these go to the National Park. They normally go to or over the West Rim which is on Indian land, and is no where near as spectacular as the South Rim NP. Besides, there is nothing like standing on the rim to look over the canyon. Try to be at the Grand Canyon for a sunset and/or sunrise. Unforgettable experiences. If you would like to camp inside the NP, get onto the NP website now, and reserve a spot as soon as you have your dates worked out.

Budgetwise you have a great trip ahead of you, but if you want more suggestions besides those linked above, check out this forum. There are also a few threads devoted to comparing the cost of car + hotel vs. RV. Those threads have some great money saving ideas as well.

Lifey