Congratulations!
It depends a lot on what journal you send it to. I would say, on average, it takes about 4 months, but others may have had very different experiences. I think most, if not all, APA journals will now let you track the progress of the manuscript online - so it tells you when it goes out to reviewers, when it comes back to the editor, etc.
Once you get it back, you'll get one of 3 responses: rejected, revise and resubmit (so not accepted but not rejected either), or accepted pending revisions. It is incredibly rare that a manuscript is accepted as is - I know of one professor who actually framed the letter from the editor for the one time she had a paper accepted without needing revision!
If your paper is accepted pending revision, the changes are usually fairly small - maybe adding more info to the methods, expanding on the discussion, or adding a table to clarify. If you are asked to revise and resubmit, the changes are typically much more substantial. You may be required to make substantial changes to your conceptual model and rerun analyses. If the paper is rejected, it is usually because either the reviewers found multiple flaws that they didn't think could be easily fixed, or because the topic of the paper wasn't deemed important enough by the editor (this can happen with some of the bigger journals).
Good luck!