I got an email yesterday from a student of a former colleague of mine who is trying to get her first novel published and finding it difficult; she asked me if I'd look at it and give her some advice. This is just an email I wrote off the top of my head, I haven't done any footwork on small presses that will take unagented submissions or the current state of play on electronic self-publishing, but I thought I might post it here in case it is useful to others. If anyone has further advice or details re: any of these options, please feel free to contribute in the comments.
I really don't have time to look at it, but can give some advice regardless!
There is no doubt it is a very, very hard thing to get a first novel published - I think it is the hardest thing I have ever done, much harder (for me at least) than getting a PhD and really distinctly harder even than getting a tenure-track job in academia, which is also pretty darn hard...
You really have 3 routes available to you, with the first being the most appealing, the second with its own advantages, the third with the least connection to traditional forms of publishing but also a legitimate route in many respects.
(1) Get an agent. Really almost all publishers, barring academic presses and a handful of good indies, want to see agented submissions and are using agents as gatekeepers. This is very difficult, though, when one has not already published! My recommendation: contact assistants of established agents, or young starting-out agents - I think that working with a well-known agent on a first novel is mostly a pipe dream, one would often be better off with their assistant anyway (more time to attend to one's business). In short, target agents under 30 who are building client lists. You should be able to focus your submissions rather than sending at large - acknowledgments pages or author blogs will usually let you make a list of agents who have represented books that you see as being in some important respect like yours (and if you write a cover letter that says "I loved book X, which I see you represented," and book X is a relatively underheralded book by a first-time novelist, it is my guess that this will also catch the young agent's attention and make them more likely to give your material a proper look).
(2) Make a direct approach to the small number of university presses that publish fiction and a select list of indies (again, think about what their lists really are like, don't bother unless there is a good fit between your style of book and what they have been publishing in the last few years).
(3) Explore some of the new forms of self-publishing or what I would call 'assisted self-publishing' that have sprung up in recent years. It really is a different world now: you can self-publish online for instance and build a readership and some online reviews and then use that to make another stab at getting representation and a traditional publishing deal. My friend Richard Nash's Cursor project would be one place to look, but there are a lot of ways to make a decent book (I'm not expert on this, I'm just vaguely recalling names, but Lulu, a couple options at Amazon, a new program called PressBooks, all sorts of other options). I do think that at this point a traditional deal is still more desirable than online self-publication unless you are an exceptionally gifted self-publicist with considerable time to spare - division of labor is a good thing! - and I caution you that if you are teaching full-time and perhaps writing another book, it will be difficult to find the energy to spare to get this one out into the world yourself. Often first books are in fact published second, after the author has written another, more obviously commercial book and gotten it published - which is to say, getting an agent to represent you is not only based on their sense of the quality of your work, it is strongly affected by their sense of what will sell, and that question might have a very different answer if you have already published something else that found a good readership.
Anyway, I hope this is useful, and I wish you all the best.