There are a lot of options here. I'll just list a few to get you started, but the best resource is probably Erik Molvar's very excellent hiking guide to the park. For fishing, Russ Schneider's fishing guide is very helpful as well.
But also know that Glacier is not really a fishing park. You can do a lot of that here, but it's rarely the fishing destination that even a few places nearby are. Honestly, I find that I have to make a decision between the two--better hiking or better fishing (in fact, it's a decision I'm trying to make for this weekend--get to a snowy pass or try a new rod).
Keep in mind that logistics in Glacier can be a real challenge. Make sure you understand where you're beginning and ending. Also, don't rely on the park shuttles to be timely. They're nice, but they're anything but on time (something that's nearly impossible with tourists involved), and construction delays on the Going-to-the-Sun Road only make this worse.
1. The Belly River - Beautiful, low-elevation country with two of the highest mountains in the park rising up from the meadows, lots of lakes and easy logistics in and out. Also very popular and hard to get into. Take Lee Ridge either in or out (in is probably easier, though out is a better aesthetic experience, in my opinion). This little visited spot shouldn't be left off your itinerary in the Belly.
2. The North Fork drainages - Lots of options here. If you want to fish, Grace Lake above Logging Lake is considered some of the best in the park. But because these lakes all end up butting against the cliffs of the divide, there are few longer trip options. The trails also tend to be brushier and have fewer views. The plus side is that they are often easier to get into (except Bowman and Kintla, which are very popular).
3. Two Medicine to Walton - This is the only really good hiking and fishing trip that I can come up with right now. The tradeoff is that you'll have both, but not at the same time. Go up to Cobalt Lake and over Two Medicine Pass for some gorgeous hiking. Stop at Lake Isabel for some great fishing. Then hike 15 miles through the woods to get to the Walton Ranger station. Or, you could go back up and over the pass (this might be the best option for views and logistics).
4. Another good option might be to go from St. Mary to Red Eagle Lake, up and over Triple-Divide Pass, and into Cut Bank. That just might make your mileage work. I heard from a guy I was talking to recently while waiting for a backcountry permit that Red Eagle Creek is good fishing; though, I can't say how reliable that information is. You might have some options in Cut Bank as well, but check the fishing guide because I haven't made it into that area for fishing--only hiking. Also, this would be a real challenge logistically, but you could make it work.