Riding south on 21st street, it will turn into Five Rocks Rd. Then turn right onto Country Club Rd and at the end of that you'll find a nice bike path to your right, running around the perimeter of the park with great views of the rocks.
Turning left you'll enter the park and wind your way toward the main entrance and the Summit Dr. Bicycles are only permitted on that road when the gates are closed to cars. Before 9 AM and after 5 PM. I arrived too late to ride all the way to the top, but did manage to get about 1/3 of the way up, promising the ranger I'd be off the road by 9.
Near the gate and to the west, you'll see Mitchell Pass and the Oregon Trail. Paved to ease your viewing.
You'll need to buy a park pass, good for seven days. $3 if you arrive by bicycle, $5 if by internal combustion engine.
Retuning to the truck, I saw another bike path running alongside the north bank of the North Platte River. This is where the Mormons travelled, creating the Mormon Trail.