Don't confuse this with the old jeep road that goes by the same name. This trail is mostly singletrack and slickrock and designed for intermediate mountain bikers.
This is a great introduction to Moab riding: the climbs are very gentle, the copious slickrock is smooth and not as ledgy as most areas, and the sand traps are short and easy to walk.
Start at the enormous parking lot ringed with red boulders (there is camping here, but there are better spots on the nearby roads). Follow the main dirt road south (the same one you used to get here) just a short ways and you will come to an intersection with signs indicating "Mill Road" and "Dino Trail" to the right. You will exit this way, so for now, stay left on the main road.
In another tenth of a mile, you'll come to a sign saying "Halfway Stage Station" to the right. Take this right turn on a very sandy, smaller road. You might have to walk it 50 yards. You are now at the historic Halfway Stage House, and the beginning of the singletrack, through a gate straight ahead.
The singletrack starts with a few short sandy spots but quickly changes to slickrock. You climb very gently up remarkably smooth slickrock following white painted markers. Once you reach the end of the enormous bluff you have been hugging to the right, there is a shortcut trail branching to the right. This cuts off the most fun slickrock, so the shortcut not recommended.
Continue riding an enormous expanse of easy slickrock around another huge bluff, and eventually turn back north. You now begin a gradual descent of the mellow slickrock and you can go quite fast. It's very smooth with no surprises, but rock is harder than dirt, so watch it.
The slickrock ends where you intersect with an old jeep road that is also called Monitor Merrimac. Go right and you're soon back on singletrack following a stream bed wash. After a mile or so, you pop out at the end of a road and parking area.
Put down your bike for a minute and take the 150 yard, signposted Dinosaur trail. There really are purple-ish dinosaur fossils and petrified wood (including a cool 2 foot diameter tree trunk!) that you can even touch.
Hop back on your bike for the dirt road ride back to the parking lot, a couple tenths of a mile away.