1) Pick a journal in advance (before or during the writing phase!)
I'll admit that I don't always do this and sometimes change my mind part way through the writing process. However, it can help guide your writing. If you're familiar with the aims of your journal of choice, it can be easier to formulate an appropriate paper. That being said, familiarize yourself with the "aims and goals" of journals you may submit to. This is most often posted on the main journal page. You can also access and read several papers from said journal. Are any of the published papers similar to yours? do they have similar goals? If the answer is yes, you have likely chosen an appropriate journal. You may also wish to ask for input from your adviser, they are frequently more familiar with a larger number of journals than their graduate students.
2) Get as much feedback as possible
You'll likely receive comments and feedback from your adviser (let's hope!). But it's also good practice to ask for feedback from others. These people might be other experts in your field (that you feel comfortable asking!) or lab mates. Hopefully, receiving as much feedback as possible will help identify issues with grammar, gaps in logic, and other errors (thus reducing the burden of reviewers!).
3) Download the author guidelines (preferably during the writing stage) and prepare your manuscript file for submission
Again, I am guilty of writing first and asking questions second (it's not a good habit!). Journal articles can be persnickety about font type, line spacing, line numbering, title page format, reference format etc. (although they may vary in specificity). This stage is usually more time consuming than difficult but follow ALL the guidelines. If you do not follow these guidelines, you risk having your paper returned before it is even sent for review (help your paper succeed!). You'll also save time. I've spent too many hours formatting and re-writing to meet journal requirements. Save yourself!
5) Prepare high quality figures/tables
The author guidelines for all peer reviewed journals should outline the ideal quality of figures including font types, font sizes, line widths, colors, image sizes, and resolutions. Most journals require that figures be submitted at sizes of 1, 1.5, or 2 column widths. Consider how large you wish the final print figure to be (will the detail be visible if the size is too small? does the figure need to be 2 columns width to be understood by the reader?) Similarly, most journals require a minimum of 300 to 600 dpi resolutions for most figures. I use Adobe Photoshop to to edit final image sizes and resolutions although other, cheaper software will also do the trick. Be certain to submit your figures in the preferred format (.jpg, .psd,.gif etc.). Graphic design of figures is beyond the scope of this post. There are numerous online and print sources on graphic design. I also recommend scientific design courses and/or conference workshops.
Journals also usually have a preference regarding table format (where to include and no include lines). As a general rule, vertical lines are not used and horizontal lines are limited to above/below the table contents. Whether the tables can be submitted as .xls files or .doc files also varies from journal to journal. Be sure to check with the journal guidelines.
6) Write a cover letter
A cover letter should address the handling editor for your journal of choice. If you do not know who it is, address it to "Editor, Journal of Unicorn Biology" or the editor of a journal that, you know, actually exists. A good cover letter should VERY BRIEFLY describe what you did, why it is unique, and why the journal should publish your paper. A cover letter probably shouldn't be longer than 1-1.5 pages. If you have never written a cover letter, send it to your adviser for feedback and ask fellow graduate students for samples from their own work.
7) Create an account and fill out the form
This is self explanatory. Most (all???) journals now have an online submission system that is usually accessed from the main journal page (buttons usually say "submit an article"). Gather all information for co-authors including their emails, affiliations, and institutional addresses or you'll be scrambling to find it at the last minute. The specifics of the online form vary from journal to journal but have some commonalities. You will be required to input author information, the article title, the article abstract, and to upload your cover letter, manuscript file, and figure/table file(s). You are usually also required to declare no conflict of interest (and if there is conflict of interest, to disclose it), to declare that all authors have agreed to submission, and to declare that the article has not been submitted/published elsewhere. Be sure that you can truthfully make these declarations (or terrible things will happen, like being bitten by hundreds of angry ferrets!). In all seriousness, your paper will not be considered for publication.
Once you upload your files, you will be prompted to build a PDF file (at least this has been the case for all journals I have submitted to, see REJECTION! http://evolbiology.blogspot.ca/2013/08/rejection.html for proof of my experience with submission). You will then have to view and approve the final PDF. Ensure that all figures, tables, and text look as they should. If not, return to the upload page and replace the offending files. Once you're PDF looks acceptable, you usually have to click approve and then submit. Don't get too nervous!
8) Wait for the results!
Most journal webpages allow you to check the status of your paper. The status will change to "in review" once the manuscript has been sent out for peer review. As a general rule, if the status does not change for three months, I will email the handling editor. I have heard horror stories of some people waiting 8 months to a year. Don't do it! It's in your best interest to make sure your paper is not forgotten or stuck on a reviewer's desk somewhere. A polite email will often get the ball rolling or at least assure you that your paper hasn't been forgotten.
Any additional advice on paper submission? Post it in the comments.