Awesome fly fishing locations in the Bozeman/Yellowstone area.......well there's.......everywhere. Really. For a fly-shop in Bozeman, I would go to MT Troutfitters. In the middle of Bozeman, and they have the best info IMHO. There are many shops in the area, but they have put me in the right spot consistently. Most shops, if you ask where to fish x hatch, they'll tell give you a 10 mile stretch of river. Troutfitters has put me right on bugs with specific spots. The Yellowstone, Gallatin, and Madison are Blue Ribbon(MT's classification for the best fishing). The Ruby by Virginia and Nevada City(ghost towns that are well worth visiting) is red ribbon. And if you fish the Ruby you can do your part to stick it to an ilk of the Wilkes brothers, James Cox Kennedy, who would like nothing more than to keep you off the river.
The 'Stone above Livingston is where you would want to fish from there to the Park. Though, skipping Yankee Jim Canyon if the water is hauling. The river tends to get blown out easily if there's any sort of precipitation upstream. And wind. You have to deal with the wind. Big hatch is the Mother's Day Caddis hatch around mother's day. Clouds of caddis. There's also the spring creeks in the area if you want to pay to fish. Go to George Anderson's fly shop in Livingston. They are friendly and I like them best. Dan Bailey's is WORLD famous. So you can go there too.
The Maddy is a fantastic and very popular river. Below Ennis lake is refered to as the lower Madison. Because the dam is a dam that feeds the river from the top of the lake, the water warms up quickly in the spring and it means its still fishable in runnoff. It also means that as the summer progresses, it becomes a place that gets too warm to fish. And you have to deal with the floaters when the water is warm enough to float(usually by mid/late May. And there are ALOT of tubers every weekend. My favorite hatch: the height of tube floating season, the bikini hatch.

Above Ennis lake is the Upper Madison and it was mentioned earlier, Ennis is a fly-fishing mecca. I like to eat at the mexican place attached to the G bar(Gravel bar). Willy's Distillery is in Ennis and has some popular booze in their tasting room. Ennis is kind of a cool town to walk around. Downtown is only a few blocks but it has a cool feel to it. One fly-shop I won't go in because if you aren't there to spend a bunch of money on gear, they won't talk to you. Literally have been ignored at the register as I was trying to buy flies. The Orvis shop in Ennis is the place to go. The guy in there is awesome. He's given me a free fly or two. There are TONS of fishing accesses above Ennis. And there is a TON of traffic at certain times in the summer. I have had people tell me you can darn near walk across the river stepping on drift boats. But it is the Madison River. And it is famous for a reason. There are few hatches that are equally renown in the U.S. as the salmon fly hatch in June on the Madison. Ask a fly shop about the Big Bugs and they know what you're talking about. Pay attention to the regs because sections of the river have different rules(no floating, or closed at certain times of the year). Between Quake and Hebgen lakes is a fantastic stretch of water. The water is always clear above Cabin Creek. If the snow is melting, Cabin Creek lets sediment into the river and cloudy's it. Bob Jacklin, a fly-shop owner in West(and who's fly shop I would suggest going to in West Yellowstone) caught a huge cutty in this stretch a few years ago that was in the paper. Bud Lilly's fly-shop is pretty well-know too. This is sitting in the mountains fishing. The rest of the Upper is in a big plains valley. Quake lake is a cool place to explore as well. If you don't know what it is, in '59 there was an earthquake that caused the side of a mountain to slide into the Madison River, and buried people camping in the area, damming up the river creating Quake lake. There is a little visitor center there with a seismograph and info. Kind of crazy to see how many earthquakes there are DAILY in the area.
The Gallatin, I would say from Belgrade upstream. Down in the valley around Bozeman is wade fishing the river through loads of trees that surround the river in the middle of a flat, open valley. Up into the canyon you go into the mountains and the road parallels the river where in high water, floaters float the rapids in spring. By summer, the whole thing is wade fish-able. Once you get up to Big Sky, the river mellows out and is flatter water all the way to the Park. There are all sorts of places to fish in the streams coming into the Gallatin as well. Taylor's fork could be a smaller stream to fish that you could share with the bears and dude ranch up the road. Hyalite Creek up Hyalite canyon is some good fishing after it opens the 3rd weekend in May. Above Hyalite reservoir is some cool fishing but you're crawling over logs and it can get tight, but fun and rewarding. Hyalite Canyon sees more visitors every weekend than any other forest service area in MT, so there's that to think about too.
If you want to fly-fish around Bozeman, you literally just go drive until you find a spot to fish. In the summer everyone goes out to fish. All the rivers and streams. I could drive up by Bridger bowl and probably find someone fishing the creek any weekend and most evenings in the summer. Tomorrow if the weather is above 45 degrees, probably most of the pullouts in Gallatin canyon will have a car. You just can't go wrong anywhere. And I'm not saying that like you're not going to find a place to fish because its crowded. Because its really not crowded. There's just a lot of people that fish here. You just have to park and walk up-stream a little. Wave to the guy fishing as you walk by. But plenty of space.