The, let's say "japanese" way, wich includes beeing a real specialist in one domain during your whole life, learning and learning again 'til you become a master (not before 75 y.o.).
Or the, let's say "french" way, wich includes working on very various projects claiming that your opened vision is the key to a fresh, flexible way of thinking (but some call it ignorance).
So, what's your position?
You can ask a question in the comments section here.
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Great question to start with. Bloody good question. I'm not 100% sure that I'm fully qualified to answer at the tender age of 31. But here goes.
I'll answer this as if you're thinking of embarking on a career in graphic design right now. Not five years ago and not in five years time. I'm sure the industry will change in the next five years.
Short answer: I would say the "French" way. Work on very various, varied projects.
Long answer: First off get a degree in Graphic Design. Why? Because you'll have fun, because you'll get a lot of stuff out of your system and because you be taught stuff like kerning and leading and picas and proportions and other stuff that's more important than Photoshop, InDesign and Illustrator.
Then get a graphic design job. Any job. Just get one, they're hard to come by. It's a cliché but it's true that you'll learn more in the first 6 months of employment than you will in 3 years of college. This isn't a bad thing, enjoy it.
Next, start moving around, don't get stuck early on working in publishing or pharmaceuticals because the pay is better than your mates get in the record industry. Work hard, be nice, talk to people. Talk to people who aren't designers. Talk to clients. A lot of my success in my first job was because I was "OK to take to client meetings", that's a valuable skill.
Gradually, depending on how talented you are and how hard you work, you will get better jobs with better companies. Do some great work that people will take notice of. Have points of view that people will take notice of. And then one day...
One day, you'll have to start up on your own. Make no bones about this, any decent designer or any designer you admire, they've got (or like Jonathan Ive and lots of Pentagram, they had) their own studio. And this is a key reason why you should work on various projects and immerse yourself in various situations, because one day it will be just you, a mouse and a blank sheet of paper.
So why not specialise? Because, these days what would you specialise in? There are a few great specialists around, but they are few and far between and I suspect they're getting rarer.
Or you could just buy a copy of Michael Johnson's excellent career tree and follow that (click on shopping).