The red line shows the trail to May Lake. It's short (1.2 miles/2 km), but climbs steadily. All that elevation will earn you some memorable scenery, though. Besides the views of the lake and Mt. Hoffmann, you'll also get cutaway views beyond the southern shore to Clouds Rest and various other High Sierra peaks (especially if you climb a little above the northern shore), and you'll be able to see Tenaya Lake from some points along the trail.
The purple line marks the trail to the top of Mt. Hoffmann. Just as you reach May Lake, the trail forks; take the left fork and it'll lead you all the way to the ridge.
The blue line marks May Lake Road (it's actually a 1.7 mile/2.7 km stretch of the Old Tioga Road), which leads from Tioga Road to the parking lot at the trailhead. The turnoff from Tioga Road is clearly signed.