If you're just learning now, you'll need close to a year of hard work before you can hope to make enough to live off of. One of the hardest parts is building a reputation and have a list of clients you can go back to for later work. There's a lot of competition too as a lot of people (especially young people who just want some extra cash, and not make a living out of it) are learning the trade and it's hard to separate your self from the pack. I suggest learning some things very well so you can sell those points rather than trying to learn everything well. Maybe you can focus on CSS or JavaScript, or PHP.
So it can be done, as others are, but it will take some time and a lot of hard work. Also, don't expect too much right away, it will take time to get good and find clients.
As far as advertising, get a site, maybe multiple ones. You'll definitely want something that you can point people too in order to show off your skills. As you do work for people you can place links on your site pointing to them. Doing some work for free for friends, family, or old co-workers can help you in working with people and learning how to understand what people want. This is especially hard if you can't meet the customer face-to-face. You'll want to make sure your communication skills are up to snuff. There is also some freelancer web sites that you can register with so people can find you. Haven't used any myself so can't recommend one.
Just to note, though not fair, people will judge you for your age. Web design is seen more as a "young mans game." Don't let that stop you though, there's plenty of ways to impress customers.