I totally agree with Dave that if seeing the Rocky Mountains is high on your list of priorities, then you'll want to head through Colorado. Farther south, say following the route of the former US-66 (which has been decommissioned and replaced by I-44/I-40 between St. Louis and Arizona), you'd be crossing the country on the High Plains and Colorado Plateau rather than crossing the Rockies. It's also worth noting that getting from St. Louis to Denver would or could entail passing through or near many locations associated with the 'Old West' if such sites are of interest to you. These might possibly include St. Joseph MO - the starting point for the Pony Express, Kansas City - the eastern terminus of both the Osage and Santa Fe Trails, Fort Scott KS - with a restored calvary fort, Dodge City - an old cattle town, and other such places.
Another advantage of taking a slightly more northern route is that after Denver, you can visit Rocky Mountain National Park as Dave suggested, then continue west on US-40 to Dinosaur National Monument, drop down on CO-139 to Grand Junction and Colorado National Monument, then continue west on I-70 over the San Rafael Swell to Arches National Park, all before continuing south on US-191/US-163 past Goosenecks State Park to Monument Valley.
Note that Monument Valley is a Navajo Tribal Park and not a National Park, so any annual National Parks pass that you would (and should!) get will not work there. But otherwise, your National Parks pass will get you and everyone in your car into pretty much every park, monument, and other member of the National Parks system. The pass does not, however, cover any 'special' fees such as parking, camping, etc. But it pretty much pays for itself if you are going to visit four or more park units during your travels. The year for which it is good starts at the time of purchase, so it makes no sense to get one ahead of time. Just buy it at the first park you come to that charges admission; they'll have the annual passes for sale at the gate.
After visiting the North Rim of the Grand Canyon, you could easily continue on to Las Vegas and then head northwest through Death Valley to US-395 and cross the Sierra Nevada on CA-120 over Tioga Pass and into Yosemite before competing your journey into San Francisco.
People can and do travel all of the above roads every day in RVs/caravans. Some roads will be slow going, but your vehicle should be able to handle them all. As a courtesy to other drivers on two-lane roads, keep half an eye on the traffic lined up behind you, and if the road does not offer many chances for them to pass you, pull over occasionally - where safe and convenient - to let them pass.
AZBuck