What training is appropriate to get to help students deal with stress? I often have undergraduate students in my office who are clearly under considerable stress.  These students are usually first years who are making the transition from high school.  This goes beyond the "school is lots of work" stress into the "I'm about to break down" level.  The university does have a counselling service that I almost always refer the students to, but they are often booked up and it takes a while to get an appointment.

I am not interested in becoming a therapist, but I would like to know some techniques for helping students through this sort of problem when they are in my office.  Things like how to get a student to calm down and how to get them to start working to solve their problems.  

What types of training should I seek out to help me with this?  I'm not even sure what the right field of study is, so I'd be starting from scratch...
It is admirable that you are trying to address this on your own, but I would recommend not trying to solve it yourself by trying to find resources. There are complex ethical and legal (like FERPA in the US) issue.

There are likely other faculty, staff, other students who find themselves in the same situation -- wanting to know what to do but not knowing where to go to learn. Graduate students also often find themselves in this position when teaching undergraduates

So why not try to address that issue for all of these groups? You say that your university has a counseling service. So why not ask that service if they would put on a one-day or half-day training course for any university member who desires to learn how to respond appropriately to students in need. My university does this, and it has been extremely successful. They even have separate sessions depending on the context in which you interact with students.