I'm finishing up a PhD in math and am thinking about options outside of academia. So far, I've really only focused on pure mathematics, but I have a year left of grad school. Suppose I am interested in looking for a job in software engineering, finance, or some other quantitative field. What should I be doing in the next year? What classes should I take?
This is the perspective of someone who went from a math PhD to the media industry and then to software/tech (and enjoyed it immensely). I can't speak for finance.
Don't spend time on classes; instead, figure out the skills you want to acquire, and learn them yourself. One important reason to sidestep classes is that from now on you will have to learn things all the time without the benefit of formal instruction! So part of what you're going to be teaching yourself is how to learn quickly outside the classroom. Also, while it's always helpful to read theoretical material, you'll probably want to spend the majority of your time learning by doing.
A good approach is to actually do the job you want. Software engineering is a very welcoming world and there are a lot of avenues. Follow and then join an open-source project. Put up your own web site that does something interesting. Help out a nonprofit. Here's a more offbeat thought: almost every newspaper and magazine in the US is currently (a) in a financial crisis; and (b) desperate for technology help. It's quite possible that, even with few qualifications, you can get an unpaid internship at a name-brand place that will net you good recommendations and excellent experience. Note that this will be helpful even if you ultimately want to work in an industry that's not desperate and in a financial crisis :-)
Finally, keep in mind that a big benefit of the "do the job you want" approach is that you'll discover whether you actually like the job before you commit full-time. If you don't like it, choose something else. And keep this attitude long after you've graduated: one of the wonderful things about the non-academic world is the total freedom to change your career when you want.
I gave this answer recently, based on a bunch of people that I personally know who went into non-academic fields. By the way, and I should mention this because I believe it's extremely relevant, these are all people based in the US (almost all are foreign-born in case you're wondering). Working in a non-academic field with a math phd is a lot more rare in my native France; your mileage may vary extremely depending on where you're trying to get a job.
The thing most of these people had in common is that they researched really carefully the career they wanted to do, and they took classes relevant for that career. It does not seem to matter what kind of math they did, and I believe most employers are not too bothered by how theoretical one is (quite the opposite, I suspect they may eye "applied" math with suspicion, as being less applicable then they think they are).
Your biggest challenge when looking for a job is selling yourself: for a non-academic job, it means really convincing your employer that your heart is in that career, that this is not a second-best for you.
Edit: to comment on your particular case further, I think you need urgently to figure out which of these fields is the best one for you, so that you spend your remaining year as efficiently as possible.