Start a blog. If you don't own a blog you can start blogging at Wordpress. Other recommended sites include blogger (it allows you to see statistics of who has been looking at your blog, along with many customization options for the layout). If you do own a blog, good for you! Go to the next step.
Pick a topic that you are passionate about. Even if you don't have a passion, writing about your daily life can make for a good read, depending how you word it. If you are going to blog about what you did today, make it interesting. People are looking for a funny story or your opinion on a debatable subject. They may not be as willing to read how you changed a lightbulb today. Using photos can help enhance the reading experience and engage readers more.
Learn about search engine optimization (SEO). The very basics of this is knowing which keywords to focus on. These are what people are searching for in the search engines, and how they might find your blog. Some of these keywords get a lot more searches than others get, so it's in your best interest to eventually choose the ones that do get searched for often. Keep in mind that these are often more competitive than ones that are searched for less -- but you might get lucky.
Focus your efforts on these keywords for now. Choose a big one that will take longer to reach, as well as three to four additional terms you're going to try to rank for. These should all be similar! Then, you're going to include these keywords in each blog post you make in various combinations. Always focus your posts on one term, and include the others only if they make sense. As you focus on these similar keywords the search engines will start to rank you more highly because your blog is tightly focused and relevant to what you're targeting.
Do what you can to get relevant links that point to your homepage and your individual posts. A lot of ranking decisions are based on how many backlinks you have coming into your website. You can get these links by writing articles to submit to directories, writing guest blog posts on other high traffic blogs, using social networking sites, using social bookmarking sites, and buying links (be very careful with this tactic).