Part One
I can respond here in sections:
I. My Background.
II. How to Use TeX.
III. What TeX is Good for.
IV. Warnings.
below:
I. My Background.
I was an early member of the TeX user's group (TUG) and have been a serious and happy TeX user for 15 years.
I use TeX itself just as Knuth wrote it and rarely have anything to do with LaTeX or any of the other extensions or preprocessors.
The TeX installation I use is ProTeX 1.2, which now is relatively old. From ProTeX 1.2, beyond TeX itself, I make heavy use of three more programs: (1) The screen preview program YAP (yet another previewer). (2) The program DVIPDFM.EXE for converting TeX output files with extension DVI (device independent file) to PDF. (3) The spell checker Aspell.
I type my input into my favorite, general-purpose, programmable editor KEdit. Nearly all my typing for anything goes into KEdit -- blog posts, e-mail, notes on cooking to car and house maintenance, programming in ASP.NET, Visual Basic .NET, C, PL/I, Fortran, Rexx, etc. And I use Aspell for essentially all my spell checking.
Point: My means of input to TeX is just this general purpose text editor. I do a lot of typing, use this editor heavily and have for 25 years, and, thus, have good facility with it. So, I minimize typing into anything else and am pleased that for TeX I can type just into this same text editor. In particular, for entering TeX I want nothing to do with any other tools for typing. I would consider converting to an editor better than KEdit, but I see no candidates.
I have a Ph.D. in applied math, and for me TeX is one of my most important professional tools. Why? A crucial reason is, for typing math TeX is well designed.
Since TeX has high quality in many respects, I also use it for nearly all my higher quality word processing -- letters, foils, technical documentation, business plans, etc.
II. How to Use TeX.
There is a book, Knuth's 'The TeXBook'. It is polished documentation, very carefully written to make first and easy things easy but still contains a lot of detail. Really you need a copy of the book and to read at least the easier parts cover to cover.
In its internal design as software, TeX makes heavy use of the idea of 'macros'. The main idea of the macros is just string substitution. The main use of the macros is just to replace some frequently used long patterns of typing with short patterns.
The macro language is powerful enough that TeX is really a programming language. Yes, the language supports arithmetic, string manipulations, if-then, loops, and reading and writing files.
The language also has an elegant 'architecture' of scope of names. So there is the nested and static scoping of what is typed in and also the nested and dynamic scoping as the macros execute. For this scoping, there is memory management with a push down stack with some enhancements.
So, a good TeX user will write some macros, and I have over 100 that I have used for years and write new ones occasionally. I have macros for tables of contents, automatic numbering of chapters, sections, tables, and figures, cross referencing, bibliographic entries, simple lists, ordered lists, and lists with bullet points, putting TeX annotation on figures, using some of the special math fonts from the AMS (American Mathematical Society), more in memory management, etc.
TeX has a concept 'verbatim' which means to produce output 'just as entered' and that might contain characters such as '$_%\' usually treated in special ways in TeX. I have several macros for verbatim, and some of these are good for listing source code and TeX macros.
To create a new document with TeX, I make use of the windows in Windows: So, I have one window with KEdit with the document I am creating, one window for the screen preview program YAP, and one window to run the TeX program itself. These three windows work well.
So, I type in my document, run TeX, and look at the output in YAP. The process works well.
If there is some tricky TeX usage in my document, then at the tricky part I repeatedly execute a fast loop of type, run TeX, and look at YAP.
TeX is so efficient that on a PC with a 1.8 GHz processor, usually TeX is done in less than two seconds, maybe less than the time it takes for me to get my finger off the Enter key.
I have enough experience with TeX that now I rarely have to look at my copy of 'The TeXBook'. E.g., my main collection of TeX macros are all in one directory and have documentation, and I have a KEdit macro that displays the directory of TeX macros so that I can easily check details. I have many examples of TeX usage, and for something tricky I've done before I just look at the old example.
Generally in my computer usage, I want to script things; TeX is easy to drive with scripts, and I do that.
Going back at least to Xerox PARC, there is a 'paradigm' of computer 'graphical user interface' (GUI) based heavily on buttons like on a kitchen appliance and a mouse for pushing the buttons. Microsoft's Word, Excel, and PowerPoint are based heavily on this paradigm. There are 'integrated development environments' (IDEs) for writing software that make heavy use of this paradigm. There are Web browsers, and now there is more with 'smart phones'.
GUIs have revolutionized computing: E.g., on the Internet GUI Web browsers are one of the main differences between the current Web which is changing the world and various 'bulletin board' systems that came before which changed little.
But GUIs have some problems; TeX is not based on GUIs, and I don't want it to be.
There is an old description: "With 'what you see is what you get' (WYSIWYG), what you see is all you've got.". To see the importance of the difference, with TeX your input is explicit, in text, in a file, and, if you wish, with some documentation of any tricky parts. E.g., my simplest verbatim macro has 54 lines of documentation and 7 lines of TeX code. So, with the 'explicitness', you know what you did, can copy it, use it again, document it, modify it, etc., and all such things are more difficult to worse with GUI WYSIWYG approaches.