I recently applied for a tenure track teaching stream position which requires to teach standard math courses but also math courses for non-science, non-math majors. It's been one month since I applied and have recieved no response. Do I just assume they went another way or should I try to follow up with an email? Also what would I ask in this email?
One month is not very much time for faculty decisions - especially if they are on the tenure track. But don't assume anything. You will probably be told of any decision no matter how it falls.
They may be actively exploring another candidate, of course, and that takes some time.
But if you have other decisions to make, then ask by all means. If you write, express your continuing interest along with asking for an update.