There are many different types of Web design careers out there, and what you choose depends mainly on two things: what your skills are and what type of environment you enjoy. Most Web designers work on a contract basis, regardless of whether they are employed by a company or hired as a freelancer. That said, there are some corporate positions available as well, but they aren't always as easy to find.
In order to get yourself on the right road, consider what you enjoy doing. Are you excited about perfecting the way something looks? Perhaps a career designing Websites would be ideal for you. If you like work that has many different components, you may want to think about working with on-line marketplaces and getting into e-commerce. On-line security is another area that you may want to look into.
Many Web designers hold a variety of different jobs over the span of their career. You may wind up starting out in one area only to find you prefer another segment of the industry. Keep your options open, as there is typically a way to transition into a different position with a little bit of training in the area. There is also usually a good deal of freelance work available.