Lots of good options.
1) The Wild Goose Island overlook (near Rising Sun, right off GTTS road) - a very popular sunrise/sunset photo spot; you won't be alone, but one of the great views for sure. Don't miss this one.
2) Logan Pass is amazing, but the parking lot fills up quick. Reasonable hike up to Hidden Lake from there. Lots of chances to see mountain goats here. The highline trail starts across the road from the parking lot, and is long and at points, quite scary (drop off off the cliff, long way down, right on the edge), but even though you're not likely to do that, at least walk the beginning part of it - great views until you get to the sketchy stuff, where you can turn around.
3) Swing over to the Many Glacier area - really nice shot at the end of Lake Sherburne and the whole area around Switftcurrent Lake is stunning.
4) Two Medicine lake area is nice.
5) If you have the time, as the other poster mentioned, Bowman Lake is amazing. I haven't been back there in a lotta years, but back in the early 90's, the road to get there was essentially semi-graded rocks - brutal hell on cars. I don't know if they've improved it since then. Gorgeous late afternoon/early evening spot with a soft, mellow light that you won't ever forget. It's a bit of a haul though - the aforementioned road is off a north/south running route that intersects with the (rough) terminus of GTTS on the western side. May not be something you've got the time (or the vehicle) for.
6) Bears are to Glacier as summer heat is to Death Valley. Take the warnings seriously, and get bear spray if you're doing any, even short, hiking.
7) Construction this year is on the western half of GTTS road only, so if you're only dealing with up and back to Logan Pass, you won't run into any.
In general, and relatively speaking, I have a personal preference for the eastern half of Glacier. However, if you choose to hop over the hill to the western side, Avalanche Lake is worth the reasonable hike for sure, and if by any chance you can end up introducing yourself to Glacier by driving the GTTS run from west to east, that's the most "amazing" way - with each successive turnout being better than the previous, even when you thought the last one couldn't be topped, until you get to Logan Pass, which to me is just staggering in beauty.