Blackboard bold, Bold, Fraktur, and Reserved Variable.

As Robert Soupe says, either $\mathbf{R}$ or $\Bbb{R}$ is unlikely to confuse a reader without explanation.

Until the last decade or two of the 20th Century, $\mathbf{R}$ was common notation for the real numbers in print journals. As Raclette notes, $\Bbb{R}$ is called blackboard boldface because lecturers would put a double spine on the "R" to make it look boldface in chalk.

Once (La)TeX came into wide use among mathematical authors (by the mid-1990s), blackboard bold fonts were, predictably, created to duplicate the visual effect of blackboard writing. Young authors (and some older ones) seem to prefer blackboard bold to plain bold for the standard number systems (and/or they pick it up by osmosis on web sites such as this one), and the notation is now entrenched.

In terms of practicalities of LaTeX coding, the right answer is: Abstract your notation, including font choices, into the preamble, using semantic macros. (That is, make the code of your document body reflect mathematical meaning, not typographical appearance.)

For example, put commands such as \newcommand{\Number}[1]{\mathbf{#1}} \newcommand{\R}{\Number{R}} into the preamble of your LaTeX file, and use \R in the body. That way, you (or a journal editor) can change your notation to match house style in a trivial, predictable way.

You can even do a cheap version of this in your MSE posts by putting $\newcommand{\R}{\mathbf{R}}$ at the start of your post, and using \R in the body.

The proper notation for the set of all real numbers is either $\mathbb{R}$ or $\textbf{R}$. It really comes down to your choice, and whichever you choose you can back with plenty of precedent. But most of the time people will understand what you mean without you having to explain it.

The problem with $R$ is that in the context of algebraic number theory it probably makes you think of an arbitrary ring or domain; in general people might be confused and think they skipped over your definition of a variable or a function.

If you're going to use $\mathfrak{R}$ to mean the reals you really need to say so at the beginning of the paper or book.

Either $\mathbb R$ or $\textbf R$ is good. The choice depends on your answer to this question: do you like blackboard bold from your aesthetic perspective?

Don't use $R$ because it's commonly used to mean a particular ring at hand, e.g., if you're discussing primes in $\mathbb Z[\sqrt{15}]$ (or $\textbf Z[\sqrt{15}]$), then you say $R$ is that ring.

I wish someone had mentioned back in 2014 that $\mathfrak R$ looks a lot like $\Re$, if not exactly the same. The latter is a function that gives you the real part of a complex number, e.g., $\Re(2 + i) = 2$. Correspondingly $\Im(2 + i) = 1$ (or $i$?), so it's probably a good idea not to use either of those to mean other things.