From the obvious trailhead at the south end of the road leading into Illecillewaet Campground, follow signage leading you 3 miles along a trail to Abbott Ridge, passing, among other things, the remnants of the century-old Glacier House and Marion Lake, along the way. Coming out of the forest, continue through subalpine meadows and eventually gain the north end of Abbott Ridge via the good trail. Here, the trail bends south and traverses the crest of Abbott Ridge for 1/4 mile or so before ending at a sign declaring the end of the trail.
From the end of the maintained trail, continue south along the crest, scrambling easily and following the many bits of use trail found here and there. Mount Abbott will be in the near distance at the south end of the ridge, with Mount Afton to its immediate west. The two peaks are separated by a snow and rock col.
Continue along Abbott Ridge, soon dropping down a short bit to a saddle between the ridge and Mount Abbott. From there, easily surmount Mount Abbott via class 2 scrambling up its north ridge.
During typical summer conditions, normal hiking gear should be sufficient. Crampons/axe may be needed if snow is present.