I am not based in academia so unsure what the protocols (and the underlying and unwritten conventions are) in regard to getting my first paper published in a peer reviewed journal.
The paper will come out of my PhD dissertation.
How do I start the ball rolling?
The procedural parts are pretty simple:
It may sound trite putting it that way, but journals are typically pretty good at being clear about what they want, both in terms of content areas and format & typography.
What you also need to do is identify what you want to take from your PhD, and determine what the "narrative" structure of that material is - a paper that's just a bunch of random things with no through-line is not a great paper.
You may also want to take the time to look at the editorial board of the journal you're interested in, find who is knowledgeable in the topic of your paper and be sure to indicate (in the manner proscribed in the instructions for authors) that you'd like them to be the editor in charge of your paper. A good choice here can help ensure that it's reviewed by the right people for the job.
Luke Mathieson covered all the necessary points. I am just adding a few supplementary ones. As a PhD student, I can assure you that opinion of your guide regarding the choice of the journal also matters (I mean, he/she wants to send somewhere and you somewhere else can create conflict and better avoided). If you have followed up works of some earlier works, then you can try in the journals where the those earlier papers were published.
One more point. Sometimes it is important to get a quick publication. For which you need to sacrifice a little and go for a journal with fast publication. You did not write your field. In Mathematics and some Theoretical Computer Science journals take an year or more for publication. TCS conference proceeding are faster and many of them are referred (and something like FOCS publication can change your life). Physics journals (theory) are generally fast. To be in safer side, keep a copy in arxiv.
