Best (as of 2019) way to install LaTeX

In the end "best" is judgmental. There are two options:

• Install a fully functional local TeX system. For that the obvious candidates would be MiKTeX, MacTeX, or TeXlive, and in terms of ease of installation for most students either MiKTeX (Windows) or MacTeX (OSX). (Linux users would probably want TeXlive, and may need to be warned about the tendency for distribution packages to be considerably out of date.)
• Use a cloud-based system, of which the most commonly used is now Overleaf

Although there are pros and cons to all these choices, there is a lot to be said for beginners in using a cloud-based approach in terms of your criteria. It offers a straightforward LaTeX compiler and editor, with nothing to install, and it works pretty intuitively. It's free for personal use. It's "real LaTeX" and mostly any document that compiles there will compile on other systems in the same way. There are good reasons for experienced users to prefer to maintain a local system and their are occasional disadvantages to being in the cloud: but as a way to get going quickly without installing anything, it may have much to say for it.

To my mind the main practical downsides are (1) occasionally the cloud-based systems lag behind bleeding edge TeX development (but that is unlikely to be a problem for most users) and (2) Overleaf seems to encourage the use of non-standard "templates", some of which are of dubious quality or utility, and which may not be portable. But (1) is unlikely to bite new users, mostly, and (2) can be avoided by encouraging people to use Standard templates.

There are obviously more principled reasons why people might refuse to use cloud-based systems and circumstances (e.g. confidentiality concerns) in which they would not be appropriate. But if your students are not handling sensitive personal data, or commercial or government secrets, and do not have rooted objections to the use of anything in the cloud, those do not seem to be deal killers. Some might have other personal views (for instance preferring to use a particular editor).

In the long run, if someone is going to use LaTeX a lot, there are very sound reasons to move away from such systems to something more controllable. At that point the sort of information provided at the LaTeX project page and by TUG is useful.

Arguably as important (and here again the cloud based systems tend to help) is encouraging users to use modern practices (such as UTF encoding) and packages from the start.

Modern LaTeX installations boil down to various attempts to make the software usable without making it larger than an operating system. TeX Live is wonderful, but it's over 3 GB to download, taking too much space on a computer with a 128 GB solid-state drive, as many of my students are stuck with. It installs the manual for every known package, even though most people probably use the online copies, especially with the advent of TeXdoc Online.

This is what I've recommended for individual installations:

• Mac: MacTeX is TeX Live for macOS; it's available as an installer or through the Homebrew package manager. There are three variants: the full MacTeX (brew cask install mactex), a slightly smaller version without the extra applications (brew cask install mactex-no-gui), and BasicTeX, a minimal installation (brew cask install basictex).

  Homebrew mostly downloads MacTeX and runs the installer for you, but it's valuable because it ensures the most recent dependencies are installed (e.g. using the most recent Ghostscript without duplicating it), and it makes uninstalling MacTeX much easier, which is especially useful when it comes time to update it every year. (You otherwise have to remove the old version of MacTeX by hand when you install a new release.)

  The full MacTeX is easiest to use, since you won't need to install extra packages, but space considerations often make it impractical. I usually install BasicTeX: you'll likely need a bit more than the package provides: I would glance over the collection packages in the TeX Live Manager (which comes with the full MacTeX or can be installed with brew cask install tex-live-utility) to see what you'll likely need. Running tlmgr install collection-fontsrecommended collection-latexextra collection-luatex to start, along with any languages relevant to your work, will take care of most missing package warnings.

  Whichever way you install it, it's often a good idea to set up MacTeX so that admin rights aren't necessary.

• Linux: Most Linux distributions have their own version of TeX Live, though they are often a year behind. There's usually both a slimmed-down version equivalent to BasicTeX and a full installation.

• Windows: MikTeX is still reliable, and can install packages on demand.

• R (RStudio): for users of LaTeX via R, the TinyTeX distribution will install packages on the fly.

• iOS: Texpad lets you write LaTeX on an iPad and has its own package manager.

• Android: See How to install TeX on Android phones?

Whichever way you install LaTeX and its packages, it's a good idea to keep things updated (for TeX Live, run tlmgr update --self --all --reinstall-forcibly-removed periodically). TeX Live tends to get far more reliable over the course of the year as packages are updated, and LaTeX itself has received some great mid-year updates of late (see the official LaTeX News).

The flip side of 'modern' is that it LaTeX doesn't have to be on one's own computer. Overleaf or Authorea can be great solutions in certain situations.