—This question is closely related to the question: ‘How do I choose a typeface?’—

I am convinced that you can only make a good choice if you know what you are looking at.

First step: context
Knowledge about history, makers, context, purpose, and concept is essential. The best way to do this is just to open up your eyes. If you see an interesting typeface, try to find everything about it. Start with the name, the foundry, and the designer.

If you don’t know the name; there are multiple ways to find out:
What the Font
Fontsquirrel
What font is

Usually, there is a clear reason for a type designer. You can imagine that the design process deals with specific research. Reading those articles is very helpful, you will be surprised at what you will find.

Peter Bilak (one of my former teachers at the KABK and founder of Typotheque) for instance, asked the question if it would be possible to design a ‘conceptual typeface’.
If you read this article, you will learn about history and context, but the most important thing: you will learn to see things.

Second step: theory and practice
One of the smartest and useful exercises I did at the master Type&Media, was something called ‘Typecooker’, developed by Erik van Blokland. I find it brilliant because it asks you to solve sometimes impossible problems. The only way you can solve those is to apply theory and resourcefulness.

So, on one hand, it teaches you type design principles like applying different types of contrast, stems, weight, width, etc.

On the other hand, it teaches you to come up with solutions that should not be possible, which might help you to get new ideas. You learn how to think out of the box.

And don’t forget the technical skill of drawing type in a fast and effective way. Within the parameters, you still have a lot of freedom.
By refreshing the webpage, you get new and different parameters. Some results you can see here.

This process of emerging yourself in the world of type by making it contextual and practical, will teach you how to look at typefaces and consequently, how to choose them and to judge them.

So this is the shortest answer, but you have to have the longest breath.