I would recommend plonk or over the tapas for dinner in Bozeman. For breakfast you must go to the restaurant in the bottom of the Bozeman Hotel. Killer huevos rancheros. Ask for the salsa that they make there. You'll get about a tablespoon, but that's all you need. Great food and great service. I wish I could remember the name of the place. I've only eaten there about 20 times.
First, you might do some searches in "Elsewhere in America" which is the catch-all including Montana.
Second, I am not a Bozeman resident, but grew up there and visit relatively often. Your first question covers "basic and good", but your second request seeks stuff with a more local flair. Those generally aren't the same in MT -- anything with huckleberry is going to be considered a little fancier and for tourists or a special night out.
Bozeman, being a college town, has a lot of basic and good, but generally without local flair. My preferences are Mackenzie River pizza (more trendy with MT flair), Columbo's pizza (basic and non-MT), Zydeco Cafe (NO/Cajun), Burger Bob's (burgers), Plonk (trendy wine bar with small plates), Over the Tapas (tapas), Pork Chop Johns (pork burgers). These places are mostly downtown.
If you're looking for a little fancier with a MT flair, in Bozeman itself I would say Boodles will be the closest you'll find. You might also try the Mint in Belgrade (10 miles away), Land of Magic steakhouse (15 miles away) and Sir Scott's Oasis steakhouse (20 miles away). Unfortunately, the place that would most do what you're looking for was Rainbow Ranch, about 60 minutes from Bozeman past Big Sky, but it just burnt down.
The best restaurant in Missoula is the Red Bird. They have a wine bar with a separate menu; I'm not talking about the wine bar. You'll need reservations. Here's the website: http://www.redbirdrestaurant.com/
If this isn't what you're looking for, what kinds of food would you prefer?