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The most immediate difference I noticed in design school was a Typography assignment. We were tasked with going out and looking at all kinds of signs and identifying what fonts the signs used. We had to collect 10 examples of each style of font (Sans, Serif, Slab, Transitional, Display, etc) and then identify the font and give critical analysis on each. How readable was the sign, did it convey the right message, was it appropriate for the brand / message, who was the target audience etc. etc.

This was a great exercise as it's something you can do as you walk about, or in your case, you can apply it to websites in much the same way. It gets you in the right frame of mind. By constantly being on the lookout, it starts to become ingrained, and you can start to identify fonts automatically. You judge what you like and dislike and what you would change to achieve a desired effect.

As with many things, critical constructive thinking is the key. And perseverance.