One of the most popular hikes in Glacier National Park is the Highline Trail. You access the trailhead from the Logan Pass Visitor Center area. The trail is 15.2 miles long and gains over 1,900 feet of elevation, so it is pretty strenuous if you hike the whole trail. If you want to shorten the hike, you can hike to Haystack Pass (3.6 miles) and back (about 7 miles round trip.)
Near the first part of the trail you reach the ledge on the Garden Wall that hangs high above the Going to the Sun Road. If you have a fear of heights, this part may be tough. The ledge is only 4-6 feet wide and drops off about 100 feet on one side. There is a cable attached to the wall that you can hold on to. We saw some little kids that were very scared on this part.
After this part, the views start to get amazing. If you do not get on the Highline Trail early, prepare to see crowds. The Logan Pass parking lot gets packed too. It fills by 10am most mornings in July and August! We saw less than 10 people on the trail in the morning and had to let many people pass us on the way back. That tends to slow you down, so plan plenty of time.
We had two mountain goats blocking the trail at one point! A ranger came along and had to shake a plastic bag at them to move them off of the trail. It was not wide enough to get by them otherwise. One day soon, Craig and I hope to hike the Highline Trail to the Granite Park Chalet built in 1914.