what graphic designers do to publish their new work?

[Happy Day Otter]

DeviantArt? it is a publishing site, self publishing, just upload your stuff, some people look for artists there. Generally they are looking for low price artists.
Do a website featuring your stuff?
Interview at a game publishing co?
Check out marketing jobs?
Talk to a printer, hang out a card etc at many printers?
Check on Monster for graphic Designer jobs?

You generally should not be concerned with publication, except if you are an illustrator for a book, but instead with getting credit and getting paid. Most of the time getting a job will get both done. And you might want to publish part of your portfolio, but not your entire works unless they are few. Deviant art will allow anyone to find the quality of your work on-line, and it might attract a few offers.
I would expect that you will have to be creative and flexible in getting a job. Take anything that is good.
But check out this for the job horrors:
http://www.clientcopia.com/view.php?id=1
Try to avoid that kind of mess, start with solid communication with the person or agency or company you are working with/for.