The suggested route of starting at Logan Pass>Garden Wall>Granite Park Chalet>Swiftcurrent Pass>Swiftcurrent Lodge is superb. However, the campsite at Granite Park is reserved in July-Aug for thru hikers only, which wouldn't include you. If you can score a couple nights at the chalet, then you'd have a chance to hike up to the Swiftcurrent lookout on the middle day, which offers the finest view in the Park. You would also be able to do it on your way over Swiftcurrent, which would make it one long day. The long steep descent from the pass to Swiftcurrent Lodge is a fantastic hike, but a real knee-buster. Also, the Granite Park area is the fairway of the bears, so keep your wits. In addition, the three valleys running west from Many Glacier (Grinell Glacier, Swiftcurrent, and the route to Iceberg L./Ptarmigan Pass) are prime griz habitat. I always see one when I'm in the area, usually from afar. Sometimes they'll shut down a trail due to recent bear sitings.
My second favorite route is to make the ~3500' grunt (you can sometimes hire a horse to take you if you don't have a heavy pack--~$100 one way) up from the Lake McDonald Lodge>Sperry Chalet>over Lincoln Pass and down to L. Ellen Wilson>over Gunsite Pass and down to Gunsight CG>to the highway at the Jackson Glacier bus stop (and your bus ride back to L. McDonald). The classic route is the reverse from this, starting at Jackson Gla. bus stop>L. McD. The Sperry Chalet is cool--they charge more, as they provide excellent full board (at Granite Park, the chalet has a free kitchen where you can cook up what you brought). At Granite, you'll likely be able to see a griz passing harmlessly by; at Sperry, you'll get to know mountain goats up close and personal. Sperry also affords a fine day hike up to the Glacier, which is cheap entertainment if you are spending 2 nights at the chalet.
A good third possibility is an out-and-back trip into the Belly River, accessed from a trailhead adjacent to the border station in the far north (on the way to Waterton, Alberta). It's a flat 6 miles to the Gable Cr. CG/ranger station, where two valleys converge. One heads up south toward the Ptarmigan wall (Elizabeth L. CG) , the other southwest up the Mokowanis R. to beautiful Cosley L. and beyond. My suggested itinerary would be to hike into the Belly, shoot past Gable Creek CG and stay at the foot of Elizabeth Lake the first night. Then come back to Gable and head up to the foot of Cosley Lake (or head up to Glenns Lake or even the splendid Mokowanis L. campsite--these would make for a long day, and also a long hike back out). Day 3 , back to the trailhead. Another way to do the Belly is to base out of Gable Creek, and do day hikes up each of the two valleys.
I call the Belly River an out-and-back trip, but you can also do it as a shuttle trip: Many Glacier (Swiftcurrent)>Ptarmigan Pass>Elizabeth L. (one long day--no other campsites along the way till Eliz. L.)>Gable Creek>customs station trailhead. The problem with this might be the shuttle. I don't think the shuttle bus serves that trailhead, but I could be wrong. If there is a bus, leave your car at Swiftcurrent and take the bus to the trailhead and do the trip in the reverse direction.
A completely different approach would be to stay in motels and do a series of long day hikes. Suggestions:
Stay at Swiftcurrent and do all three valleys:
1) Iceberg Lake and/or the Ptarmigan tunnel (doing both would be a long day)
2) Hike up to Swiftcurrent Pass, and then the lookout.
3) Grinell Glacier
Stay at Rising Sun Lodge west of St. Mary, shuttle to Logan Pass, and hike the Garden Wall to Granite Park, descending past the campsites to the Loop (shuttle stop), and bus back to your car.
Other good mild/moderate day hikes to fill up your time would include Avalanche Lake (scads of hikers, great destination), Hidden Lake from atop Logan Pass, and Cracker Lake in the Many Glacier/Swiftcurrent area (this has a campsite too, in case you want to spend the night).
Don't disregard the seldom-visited lakes on the NW side of the Park--Kintla, Bowman, and Logging Lakes. Though they don't offer the fantastic scenery that the others I've mentioned (not bad though), they do offer the wild Glacier experience. They all are flat trails, running along the lakes. Access to the lakes is off the rough frontage road, so 4WD is helpful, though not mandatory.
Now, about campsite permits. As an earlier poster mentioned, about half of the reservations are saved for the day before. I never count on it though, as people show up and form a line very early in the morning for the most sought-after sites. Then, be aware that even if you get a permit ahead of time, they can also revoke it in the case of bear activity. So have some backup plans in mind.
Meals in town: St. Mary--the Park Cafe, great everything, long wait; Babb (near St. Mary)--the Babb Press for good sandwiches and breakfast and the Cattle Company for a good steak.
I could go on and on about the SE wild part of the Park, the neighboring Bob Marshall wilderness area, etc. But I think maybe I've given enough info for now. If you have any questions, please ask. I grew up in that country, I love it dearly, and I want you to enjoy your trip in a most beautiful part of the planet.