I want to exchange links with sites that are relevant to my own (web and graphic design), but that doesn't make sense to me because what designer is going to want to advertise a competitor (and why would I want to)?
Advertise the heck out of your competitors if they are better than you, that's what I always say. You will gain more from exposure to their website than they will form yours. Plus, even when you are bigger, trade favors both parties. So the other guy will grow. So what? You will grow also and will be mutually benefited. If you ever study macroeconomics, it is the concept of trade and comparative advantage that essentially guarantees your success. So long as the traffic rankings are SOMEWHAT close and the topic is in focus
You are right your competition will not want to trade links with you. You should focus on building links through article submissions, blog postings, and directory submissions.