Primary image by MoapaPk; great information in the caption.
The scrambling route up the west face of Red Cap is apparently what is considered the traditional route on the peak, and it is the most direct of the three routes I have used there. There is a good deal of Class 3 terrain, some exposure, and two "easy" Class 4 spots (one of which can be bypassed with a little "cheating"-- see route details). On a hot morning, this would be a good route since you will stay in the shade for much of it. On a warm afternoon, the route is a delight.
From Charleston Boulevard (Route 159) west of I-215 and Summerlin, turn onto the Scenic Drive in RRCNCA. Just before Mile 3, you reach the signed Sandstone Quarry area (it is the third developed pullout along the road); turn off and park here.
RT distance is about 2 miles, and elevation gain is about 650’ (4280’-4925’).
About 3 minutes along the Calico Tanks Trail from its junction with the Turtlehead Peak Trail (or about 5 minutes from the parking area), you will reach a spot where the trail bends sharply to the right as it meets a wide, gravelly wash. Leave the trail and follow this wash-- avoid going left when you intersect another fairly wide wash-- through and over some small dryfalls and narrow sections that would feel like slot canyons if the walls were a bit higher. This wash will take you into open terrain near the left end of the peak’s western face. There are many ways to get there, ranging from easy to hard, but the easiest is probably this wash. The first photo below shows the wash where it meets the trail.
At the northwestern base, hike up easy sandstone slopes and get as high as you can before the slopes turn into cliffs. At that point, start traversing south (right) across the western face, aiming for a section with dryfalls and a gully. This may sound vague, but it is easy to locate on the peak, as going either too high or too low will result in cliffing out. Following the next paragraph, there is a picture showing this spot (first picture).
Climb up along the dryfalls. It is possible to climb the dryfalls themselves for some distance, but they eventually steepen into Class 5 terrain. However, there are ledges and ramps just to the right that work. They are Class 3, and arguably Class 4 in places, and somewhat exposed higher up, but they should prove easy for any experienced scrambler.
Above the dryfalls, make what the diagram on the primary image refers to as an exposed traverse to the left (north). Personally, I felt that the ramps here were sufficiently wide that I did not feel exposed, but everyone's perception of exposure differs. (It's also possible, though it seemed unlikely considering the terrain, that I did not follow the same course.) Anyway, the traverse is short and then leads one into the area seen in the second and third pictures above.
Scramble up (Class 3 at most) and reach a shallow slot, climbing out of which seems to be the key to reaching the upper part of the mountain, where the yellow sandstone changes to red. There are two viable options at this point. The first is the "cheating" option and is shown in the first photo below; wedged against a tree is a pile of rocks someone seemingly made as a step. This step makes the transition from yellow to red much easier, especially for people who are not tall. The second option is a little to the right of the step and is shown in the second picture below; it is a short Class 4 pitch that is mostly a friction climb but does have some small holds. (Full disclosure-- my first time up Red Cap, using the White Gully route, I did use the step, not actually noticing the other pitch, but I used the other option the next time I was at that spot. Anyway, if that step is still there when you climb, don't feel ashamed about using it if you judge that it is the best and safest way for you.)
Alternate Way: SP member Stu Brandel found an easier way up about 150' south of the cairn step and tree. He says it is a little bit of Class 4 at the end but easier than the wall. Please see below for two pictures depicting this option and read the captions for additional information.
Once up, hike up the red slopes in the direction of the summit until you reach the tinaja that is one of the most distinctive features of Red Cap. It is not very deep, but unless you climb in summer, which isn't recommended, there should be water in it. By the tinaja, notice a crack about 8' high (pictured below). Climb it (marginally Class 4, could get harder if small holds break off, though) and then wander a short distance counterclockwise until you reach an easy break that gives up the summit itself.