The Gallatin is great fishing, I have (share) a place in Big Sky, the fly shop Gallatin River Guides in Big Sky is great for guides, information and goodies.. If you stay in the Northern part of the park or Big Sky or as far north as Bozeman you will have access to tons of places to fish. Look at or around the small town of West Yellowstone if not Big Sky, Three fly shops in the town of West Yellowstone, Blue Ribbon flies, owned and operated by Craig Matthews is fantastic, as is Bob Jacklin's shop and Bud Lilly's shop. Take your pick. They will all have excellent up to date information and recommendations that will tell you the best options happening in real time. If you do hire a guide, which is a good idea, any of the shops have excellent guides that would be well worth the money. That area has access to streams in the park (special fishing License), and North of the Park. All great options. It also has access to the Madison (NW) which is best floated but can be wade fished from the Slide Inn down stream for quite a ways. Be careful, there are some pretty dangerous areas to wade there. Check with the Slide Inn fly shop owned and operated by Kelly Galloup for conditions and recommendations, again, an A #1 shop. The Gallatin from the Northern Park Exit all the way to Bozeman is very fishable and a great option. I know it well and can even give you specifics on where and what to use (flies) depending on time of year. West Fork of the Gallatin outside Bozeman is also good, but Stream access is tougher and tricky unless you are with a local or someone in the know. The Missouri for fishing would not be my first choice with so many better choices (tribs to the Missouri) for first and second choices. The Bitteroot may not be a bad option, but is hugely dependent on weather and stream flows, so it can be hit and miss.
I could go on for hours without specifics.
As the trip gets closer let me know and I can dial you in. Locations and dates (for flies) and stream flows.
Pictures are of a Brown and Rainbow from the Madison Downstream from the Slide Inn.