Mallory Cave (closed April 1 to October 1) is the main draw for people on Dinosaur Mountain. The trail is well-marked and well travelled. However, if you continue above the cave, you'll get to several wonderful cols and, after some adventurous routefinding, the actual, spectacular summit of Dinosaur Mountain itself.
The standard approach starts at the NCAR trailhead. Total distance is about 3 miles and takes about 2 hours. The Mallory Cave trail can be combined with the south-west side trail for an interesting tour de Dinosaur.
Hike the Roberts Interpretive trail from NCAR to the NCAR trail. Follow this past the water tank to the Mesa trail. Head southwest on the Mallory Cave trail as it slabs uphill and through a rocky draw. Continue past Square Rock and several small flatirons. The trail reaches the northeast corner of Dinosaur Rock, the first large flatiron you'll encounter, and starts a long series of short switchbacks past Der Zirkle and the northern face of Dino Rock (first stratum). Once above this, you'll bear north and reach an opening over massive boulders with excellent views of Boulder below.
The Mallory Cave itself is directly west and uphill up a steep (3rd/4th class) ramp and is well-signed as shown in the first photo below. For the summit, bear right and follow the climber trail (Box Express), as it switchbacks up a steep ravine between the Hand and Finger Flatirons and Red Devil. Soon you'll also pass the Box and emerge at a lovely, wooded col west of the box.
The third stratum is marked by the five spires of Fee, Fi, Fo, Fum, and Dum. You want to continue west and a little north and pass between Fum and Dum. It doesn't look like much, but a short scramble up some broken, jumbled boulders (see the second photo, below) will lead to an obvious and increasingly steep trail. This will drop you into the sandy 'Bowling Alley' between the third and fourth strata. I won't tell you all the secrets because half the fun is finding your own way up here. If you're climbing anything harder than third class, you're off-route.
The summit block of Dum is a great scrambling problem as well and feels much more precarious than it is. The actual summit is a few hundred feet to the east and a few feet taller. Again, you can gain it with nothing but 3rd class scrambling, but it may not be obvious. Be sure to check out the northernmost pinnacle of the fourth stratum as well for spectacular views down into Skunk Canyon.
Nothing but good shoes and standard hiking gear is needed. Bring binoculars to check out all the surrounding peaks and the view of Boulder below.