Nick, there are a lot of great rental areas in Bozeman, but I would suggest you look in the Baxter Meadows area. There are a lot of condominiums and apartment complexes in the area. If you are looking for something cheap and dog friendly, Trout Meadows LLC is a good place to start.
But you should also consider what your goals will be for finding a rental in Bozeman. Work? School? Location does matter a good amount as public transportation is limited. You may want to focus on a places that will be close to where you and your girlfriend may be working.
If you want to talk in more detail, don't hesitate to contact me.