I lived about 60 miles the eastern side of Glacier for about 10 years. I've hiked nearly every trail on that side of the park. I obviously have some favorites, but nearly everyone is amazing scenery. I've also backpacked into and camped at countless lakes on the eastern side of the park as well. The eastern side of the park used to be a lot less popular, but its now busy everywhere in the park. Make sure if you're planning on renting rooms in the park you get reservations as SOON AS POSSIBLE.....it fills up incredibly fast, even the backcountry campsites will be full really early in spring.
A couple of things to consider are the going to the sun road usually doesn't open until mid to late june, but depends on snow levels (lots already this year). The bugs (mosquitos, black flies, etc) can be absolutely maddening by 4th of july until the first hard freeze. The fly fishing in the mountain lakes is AMAZING so if you're a fly fisherman a fly rod is a must (otokomi lake is my personal favorite). The campgrounds will be full by mid-afternoon everywhere Thursday-Saturday, although still very busy other days will be much more doable later in the day.
The Many Glacier lodge is a must see. The history of it is amazing, and its very interesting looking at all the pictures and stories on the wall. I would also say that a person must hike to at least one of or both of the Chalets......they are really amazing in my opinion.
As far as wildlife viewing, you can see animals everywhere in the park but the western side is much more condusive to seeing wildlife. Black bears are plentiful and have hardly made a trip without seeing one. If you want bighorn sheep then slides and mountains (even the shoulder of the road) at Many Glacier is the ticket. Mountain goats are everywhere but the top of going-to-sun-road has lots really close to parking areas and the goat lick on southern end of park has lots of goats easily viewable. I've seen lots of moose around the St. Mary area but again can be seen around any of the lakes and rivers. I've spent many nights up at gunsight lake (on the way to Sperry Chalet, and also GREAT fishing) and the mule deer are almost tame.
If you're going to do any backpacking or back country hiking I'd say the best bang for your buck (on eastern side) would be around many glacier. There is trails up to some REAL glaciers, great vistas, the Ptarmigan Tunnel is really neat, and again the lodge is really really neat. The grizzly bears are thick in that area and often have the trails closed so cross your fingers.
Hope some of this helps. It's one of my favorite places on earth, and everyone should see it at least once. It blew me away that many people that live close by have never even been there......sad