The route Michael mentioned will not add much more than a couple of hours to your journey time, whether you decide to head to SLC or not. With 4 days from St Louis, that's not much of a difference, but you won't have too much time for side trips. I70 is one of the most scenic Interstates and has decent and interesting towns just from it, like Glenwood Springs, Grand junction and in Utah, Moab with Arches NP nearby.

Remember the 'I' stands for Interstate and although mountainous in places you will not come across narrow, winding roads. All Interstates are built to standards to take the biggest of rigs with gradual curves and gradients.

You might get close to the CO State line on day 1 from St Louis [Oakley KS perhaps] and push on to Moab next day. That would be two full days, but you could then take a relatively shorter day and end up in Ely on US50 before finishing off in the Bay area. Of course you can juggle it to your own tastes and whether or not you want to head up to SLC or not. SLC to the Bay would be an extremely long day on the road and would make it more difficult to even out your stops.