Although this route is mostly 4th class, it is an enjoyable climb and provides a scenic tour of Glacier Gorge and Taylor Peak. The climbing is similar in character to Kiener's Route, including some tiptoe ledges much like Broadway on Long's Peak. It's an easy climb in an exciting place.
At the bottom of Taylor's east face, find the hanging snowfield angling left toward the peak's southeast shoulder. Climb the snowfield (in July this was about 400 feet of crampon-friendly snow, approaching 55 degrees) to the top where it ends at a small col.
From the col, turn right and climb a short 5.6 pitch up any of several dihedrals and corners. Traverse right and up, out onto the east face. This is mostly 4th class, but you may find a short pitch or two of 5th-class where a rope is desirable.
Nearing the central buttress you will find a large, shallow gully with many short, steep steps, faces, corners, and headwalls. Climb straight up the gully on very enjoyable 4th-class ground to the prominent notch southeast of Taylor's summit. Scramble up onto the gentle west side and then up to the summit.
The easiest and most enjoyable descent is via Andrews Glacier to the northwest.
Approach as for Taylor Peak; the snowfield start will be obvious at the base of the peak.
A light alpine rack is sufficient. Some snow protection would also be useful in winter or spring.