The short answer: both. There are tons of web developers out there, and you'll be competing with overseas companies and freelancers who charge nothing. One of the ways to differentiate yourself from them is to be a full-service provider. Not only can you build a site for a company, but you can support and market it long-term.

There is plenty of recurring income when you develop a site, most of my clients have been with me for over 10 years. Web development is not a 'build it and you are done' type of thing, as the web is constantly changing. For example, sites I built 10 years ago obviously need face lifts to keep up with the current trends, and with the advent of mobile phones, responsive design is now the way to go. Some sites get redesigned yearly.

Pick up as many skills as you can so that you can offer your potential clients all of the services they need to be successful.