Write as fast as you can. The more quickly you get your ideas on paper, the more comfortable you will be with them. Don't worry about typos, spelling or grammar at this point. Simply write.
Begin with the body of the essay. The introduction to a paper is usually difficult to write. Waiting until you get the introduction perfect will only slow the process of writing the rest of the paper. You have already planned your thesis statement in the outline stage, so start writing from that point and leave the introduction for your second draft.
Mark the section of your paper where you aren't sure what to write next. Use brackets to highlight areas that you know will need extra thought. You can always go back to them later. Keep writing until you finish as much of a complete draft as possible.
Rewrite your thesis statement. As you write the draft of the paper, your ideas will become clearer to you. If there is a better way of explaining what you are writing about, it is good to rewrite the thesis statement to reflect your new line of logic.
Write your first draft in whatever way is most comfortable to you. Most students type their papers, but you can also write a draft long hand. Some students prefer white college-bound paper, while others prefer yellow legal pads.