The Highline Trail from Logan Pass is a must-do. You can basically go out and back as far as you want to. The first mile from Logan Pass is absolutely incredible (for those who aren't terrified of heights and a relatively narrow ledge). Logan Pass is also the origination of the Hidden Lake Overlook trail which is an iconic hike. The Avalanche Lake trail on the Lake McDonald side of the park is another popular moderate hike that is definitely worth the investment.
How many days are you in the area and how many days are you scheduled to be in the park? A trail ride or canoe/kayak rental at Many Glacier is definitely a worthwhile experience as well. On the Many Glacier side of the park, you can also hike up to the Ptarmigan Tunnel. It's about 10.5 miles and considered a "strenuous" hike, but it's a good way to spend a day, if you have an extra day.
If you are staying outside the park on the west side, a side trip to Hungry Horse Dam in Columbia Falls is an excellent way to fill a little time in the afternoon or after dinner. I wouldn't call it "must-see," but if you are there anyway, it's a good way to fill an hour or so of "dead time" late in the day or on a day where you don't feel like dealing with the crowds inside the park.
Where are you flying into? If you are driving up to Glacier from Missoula, a side stop at the National Bison Range is definitely worth it. It has a one-way gravel road that winds up into the highlands of the range for about 17 miles. On a good day, in addition to the bison, you'll see elk, deer, bighorn sheep, antelope, black bears, and beautiful views. On our last trip, we saw three black bears, all within about 50 feet of the road.