What's the fastest way to preview changes as I learn TeX?

Quite a few PDF viewers support SyncTex or can simply reload the document when it is modified. This is the case of Evince, Okular, among others.

I don't know what editor you're using. I'm personally used to vim, so I can just build from within it (and the PDF can be refreshed automatically as stated before).

There are also TeX specialized editors with a build button and integrated PDF viewer, so you don't need to go to command line and open the PDF yourself. TeXworks (shipped with both MikTeX and TeX Live) is one of them. On KDE, there's Kile which is also an excellent product.

As far as WYSIWYG goes, LyX is a program that wraps around LaTeX to make the whole experience more "WYSIWYG". You can use a WYSIWYG editor, or a LaTeX editor, as you wish. Afaik, it's not pure LaTeX though (it needs some converting).

I personally use latexmk with the -pvc (preview, continuously) to recompile the document every time it is saved. It also (re-)opens the PDF viewer of your choice automatically for you. This should work well under Windows, Linux and on the Mac.

Usage:

latexmk -pdf -pvc mainfilename

In addition of all the desktop solutions provided, you also have some online alternatives, which can be especially useful while learning. A couple I'm aware of are:

• Overleaf

  

• ShareLaTeX

Overleaf features instant preview with automatic refreshing, PDF export, and (with subscription) the possibility of saving your TeX in the cloud, among other interesting features (such as sharing or collaborative edition). Also it has a pretty complete TeX installation with most of the usual packages available.