I live in DEN and can recommend TONS of things to do here. But like the others, I can't really recommend going to Mt. Rushmore from here, since you're signing up for 16 hours of driving for a one hour stop.
Driving to Yellowstone from here is nearly 10 hours one way, and as one of the other folks mentioned, not real scenic until you near the Tetons.
You might consider flying to SLC instead of DEN. That cuts your one-way drive time to about six hours instead of 10, and it's usually cheaper to fly to SLC from LAX than to DEN or Jackson Hole.