Bozeman is a great spot to have access to several rivers within a 15 - 90ish minute drive. The Gallatin is the closest (15 - 60min drive from Bozeman depending on where you go) and is a great river to wade fish - it's my favorite. Anywhere from Belgrade upriver will have good water that holds fish, but the canyon below Big Sky has a lot of turnouts so you can quickly move from spot to spot. As mentioned, beware of close cars though. There's also lots accessible areas on the Gallatin where you can get away from cars if you want. In the valley between Belgrade and the start of the canyon there's some access but also quite a bit of private property to be careful you don't trespass. It's all pretty well marked though. I would say just drive South on the highway towards Big Sky from Bozeman and stop when you see an irresistible spot (you'll be able to see the river most of the way) - you might not get far your first day.
The Lower Madison (about 30 - 40 min from Bozeman) is shallow and great to wade, but can get warm in the late summer, sometimes so much that the local authorities institute a "Hoot Owl Closure" that prohibits fishing from 2pm to midnight to protect the fish. This doesn't always happen, but can be put into effect with no advance warning so be sure to check the local fishing reports each day before heading there for the latest info. The Lower can also have a lot of weeds in the river sometimes during the summer, making the fishing unproductive and not fun. There can also be a ton of inner-tube floaters on the main stretch from Blacks Ford to Warm Spring's Access Sites during the summer if the weather's good (the beer hatch) which can spoil the fishing some. I would go up Bear Trap Canyon as opposed to down river of Blacks Ford to avoid them at that time of year. Depending on the specific conditions though, it can be a great river to fish. Just check the local reports to see what's going on.
I don't get out to the Upper Madison much as it's significantly further than other options that I personally like more (an hour or two drive from Bozeman), but it's not that far and is a good river to wade, though not nearly as mellow as the Lower. The fishing can be great up there though and it's really pretty, so check the local reports to see if it's worth the trip.
I haven't fished in Yellowstone Park as it's a pretty far drive (though definitely doable in a long day), but there's some incredible water up there and it's gorgeous, so I would recommend checking out some rivers there. The Lamar Valley has quite a few good options. Heads up though - barbs, lead, and felt soles are not allowed in the park and you need a separate National Park fishing license besides your MT state one (it's not very expensive though).
The Yellowstone (outside of Yellowstone Park in Paradise Valley, 30 to 90 min. from Bozeman) should be in lower flows in late summer and wadable in spots but it is still a big river that's best fished from a drift boat. If you opt for a guided float trip, this would be a great river to do it on. I've worked with Angler's West fly shop in Emigrant a lot and they do a great job guiding people on the Yellowstone. If you want to try wading it, drive South on highway 89 from Livingston and check out the fishing access sites you can see along the way - most have some wadeable areas. Carter's Bridge, Mallard's Rest, and near where Depuy's lets into the Yellowstone might work. There's also other places much further south on the way to the park that could work. Again, it's a big river where people die every year, so be careful. Awesome river though.
