This post comes as a therapeutic exercise after having spent longer than I'd hoped bogged down writing an academic journal piece. I wanted to write a little on the topic of getting published as a postgraduate, when to start and how to go about it. I did write on this before, but this is an expansion and refresher, so you can have the topic again!
If you want a career in academia, you should already know this, but I'll say it anyway: you have to publish. In the UK the upcoming Research Excellence Framework will require new staff members to submit two publications and your department will be depending upon it. Publishing also helps build your CV, gets your name out there, shows that you are working on innovative, high-quality research and allows you to contribute to the scholarly debate in your area.
I believe, personally, that Open Access is an ethical decision that researchers should be taking so as to avoid undermining their own institutional access to publications. That said, Open Access is still gaining an Impact (shudder) foothold, particularly in the humanities, or demands unreasonable (for a postgraduate, at least) author-pays fees. With that in mind, you're aiming as high as you can on the impact metrics. If you aren't confident, then a postgraduate journal is great. I helped setup and was chief editor of Excursions and whether publishing with, or from an editorial, point of view, it's a great experience. I don't know that it will get you a job, though.
Push yourself. If your supervisors are saying your work is good, try and get it published. Journals have peer review procedures; if your work isn't good enough, the journal will let you know. Often, peer review feedback, at least in my area, tries to err on the side of positive responses aimed at improving your work. Try again, fail better. That said, rejections are part and parcel of the process -- don't expect the #1 journal to just take your publication.
Watch out for CFPs (UPenn and H-NET are good resources) and apply. Also, ear to the ground in your field: edited collections will most likely be called for by your peers at conferences. Pitch for abstracts on the fringe of your research with which you can tie in.
As already mentioned, working from an editorial standpoint can also be beneficial. This can either be with a postgraduate journal or, if you feel ready and have good, established people in your field who trust you and are also keen, you can open up to all, as I have just done with Orbit: Writing Around Pynchon.
Right, therapy over for me. Good luck!
Featured image by the trial under a CC-BY-ND license.