These are two completely different fields and require two different personalities and approaches. It would be very beneficial if you were very interested in your chosen career direction and I've never met a person that would be in both of those.
Web design requires a lot of creativity. True, you should know some server side but then the more you know about it and work in this direction the more you become a web developer and not designer. If you like images, colours, are very interested in what people like, accessibility and are arty and creative then this must be for you. You need to be able to put someones design requirements together and so they like it. If you want to do your own websites then you're more of an Internet marketer.
As a web designer you can work for yourself (freelancer) or work as an employee so you're fairly flexible. There is lots of competition out there though.
Networking (if you're referring to IT) is a completly different field. You would work for an employeer and normally to get very good at this the employer would be a big organisation. You would also need to get many certifications to be recognised (Network +, Cisco - CCNA and CCNP, security certs). To be successful in this field you have to be technical and like to resolve technical problems. You could normally progress from a different support environment like desktop support.
Hope this helps a little.