Benjamin, sadly I won't be able to join you, as I'll have to work most of those days.
You ought to know that the park still has a lot of snow, an unusual amount for this time of year, as three feet fell at the highest elevations last week. That snow melting led to a fair bit of flooding, and will make for higher than typical stream crossings when you're here. Another thing to keep in mind.
No sense in getting advanced permits now, but you should give getting a BC permit a bit of consideration before you arrive. Early July isn't busy, but a number of folks with advanced reservations will probably be scared off their planned trips by the aforementioned conditions, thus crowding the more snow-free areas. The walk-in page is your friend here: http://home.nps.gov/applications/glac/bcpermits/bcbull/bcrescgstatus.cfm
If you have three days for the park, I'd get your permit first thing. You can get one walk-in on the morning of the day before you start. Then that first day drive up to Logan Pass (which should be open by then) and spend a day dayhiking around and seeing the sights. The next day and the day after, do an overnight. Keeping to the east side of the park is simplest. A trip going out to Cobalt Lake or Upper Two Med Lake, hiking on one side of Two Med Lake on the first day and completing the loop on the other side the second, would be a fine choice. An out and back in the Belly River area, staying at Elizabeth Lake, Cosley Lake, or the foot of Glens Lake, would be another good option. Both would have a fair bit of traffic, which might allay concerns about going solo if it comes to that.
The above advice is assuming you're not comfortable with much snow climbing. If you are, say so and I'll give you some other options.
