Hiking Glacier is a treat, and to give you an idea what a treat this trail is, we hiked in early July and encountered not one but TWO crews of trail maintainance workers, who shared that "this is where they come to work when they get a chance". The trailhead is actually Medicine Grizzly Lake, then a turn uphill to Triple-Divide. The views are stunning, the trail is very lightly travelled (but easy to follow), and wildflowers, wildlife and snow (we covered several patches of snow, and ended on a snowfield) will be found.
Take rain gear, as in all mountain hiking the weather is unpredictable and we got both rain and hail on our hike. Mud will be heavy at lower elevations if there has been rain, so bring waterproof footwear and a change of socks.
We hiked this in a leisurely 8-1/2 hours, stopping for lunch and lots of pics (you'll want both 200 and 800 ASA film, and a wide-angle and telephoto are worth the extra weight--I used 96 exposures). Our daughter is 8, which slowed us a bit, but she made the whole hike, so it's not too tough.
The gem of this trail is that you hike IN as well as UP, so very soon you are surrounded by mountains and no evidence of human activity invades your world. Also consider hiking this one-way: If you camp at St. Mary's(east entrance), and get someone to drop you at Cut Bank trailhead, you can hike through Triple Divide back to St. Mary's (about 20 miles), which will take you around the lake as well.
Last note: at the end of any hike in the eastern part of the park, reward yourself with a piece of pie (my fav. is "razzleberry", a combo of huckle/blue/razz-berries)at the "Park Cafe", just north of the east entrance. It is the single best pie in the world (and I lived for awhile in Amish country...)!