Steve, I grew up on the eastern side of the state and since high school have lived in most of the towns you are interested in. I've been in Helena for 9, was in Missoula for 4, Butte for 5, and bozeman for one terrible summer.
Honestly, to each their own, but no way would I live in Bozeman, even though it is closer to home. I would live in Missoula hands down again over Bozeman. I spent my college years in Butte, so I'm extremely fond of Butte. If I was in your shoes, I would give Butte a hard look, we did when we were debating on moving to Helena or Butte when my wife finished school in Missoula. Butte has very low cost of living compared to any of the other areas you are considering and it is still very central to everything you want to do. Butte definately takes some getting used to, I hated it my first year of school, after that I loved it. It has to grow on you like the cancer the water gives you.

Helena is fine, we have lots of family here so that has been nice. My biggest complaint is since it is central to all the other bigger towns, you can't really escape the people in any direction to go hunting. Every direction you go you start bumping into guys from the next town.
My advice would be to find where there are jobs available to start with and go from there, that is a tough aspect of montana, there isn't a ton of opportunity for jobs like that.
My wife works in the medical field here in Helena and we have a lot of friends that work in medicine around montana, if you have any questions about specific hospitals, I might be able to get you some answers. Also, open invitation to hit me up if you are coming through Helena looking. Good Luck!