The first step is to start a blog. You can create your own blog dedicated to discussing your travels and experiences. You can write several blogs on the same place and just talk about different aspects. In one, you can talk about your experience at a specific hotel or about the service. In another, you can talk about the kinds of food and shops are in the area.
Do some Internet research and take some queues from existing blogs and find out how they structure their articles. You can even find a niche to specialize in if you want to go for specific audience or magazine. When you've got a good following or when you've got several articles on your site, then you can begin petitioning paying travel magazines and websites about writing articles for them.
You can use your blogs as samples. The only caveat about this is when you visit places for vacation, you have to take notes and go out of your way to find topics to write about. Some people get discouraged because it turns their vacation into work, but that's the price you pay for becoming famous. Pretty soon, you'll have your own show on Travel channel and get paid to visit places and stay in hotels.