Most of the west side of Amphitheater Mountain is open terrain where it is easy to see where to go and where not to go. There are cliffs but they are easily avoided. Any climb on the west side should not be harder than Class 2 unless you like making things more difficult.
The access gully on the north side of the mountain allows easy access to/from Cathedral Peak. Without this access gully it would be a long walk around Amphitheater's northwest side.
There are several rock climbing routes but I have no first-hand knowledge of them. If you have information, please post it here and I will incorporate it into the route page (or you can create an entirely new route page).
From the trail (7,300 ft) at the west base of the west summit, contour rightward (southeast) through sub-alpine forest aiming for the amphitheater basin west of the main summit. There are a lot of possibilities but the easiest terrain continues northeastward to the head of the basin near to the north summit (8,200+ ft). (Note the hidden access gully from Cathedral Peak tops out here.) There may even be a trail in the sometimes sandy terrain. Once up to the ridge the main summit is to the right (south) and is an easy walk. The west summit is an easy walk to the left (west). The north summit is a short walk/scramble away.
North Access Gully
This gully cannot be seen from Upper Cathedral Lake but can be seen not too far east of it. As you walk the trail to Cathedral Pass a broad "gully" will come into view. This gully is largely a scree slope leading up to a saddle. It is easy to get to the base of the slope. In early season expect 30-degree snow to reach the saddle. In late season expect loose scree. Class 2 but irritating. This gully tops out just west of the north summit. And when you get there you'll be glad you used it.
If doing Amphitheater first and then Cathedral you will just have to trust that the gully is there. It's pretty obvious when standing at the saddle looking down toward Cathedral Pass.
Technical gear will not be needed for the standard routes.
Warm weather gear is advised (it can snow at any time of year in the Pasayten).
If you have any route information--particularly concerning rock routes on the north walls--please post it here.