When I learned division in elementary school, I learned "remainder" at the same time. I think it is mostly terminological that this is not called an "operation", because the division algorithm produces both the whole number quotient and the remainder of division of two natural numbers, at the same time.
On the other hand, when we move to the rationals, "remainder of division" is no longer a very interesting operation, because the rationals are a field. Students are taught to stop using the division algorithm and start using a different algorithm to divide fractions. This is perhaps a reason that the remainder operation is de-emphasized. But students are certainly still able to compute remainders if they are asked to; they just don't describe it as an "arithmetical operation".
Well, since parentheses exist precisely to specify the intended order of operations in case the usual default rules don't cut it, it makes sense that they come first
As for exponentation, I'd say that this is a consequence of using superscripts to indicate exponentation, since those (via font size) provide a natural grouping. It'd certainly be very weird if $a^b + c$ meant $a^{(b+c)}$ instead of $(a^b) + c$, since the different font sizes of $b$ and $c$ indicate that they're somehow on different levels.
As MJD pointed out though, this arguments only applies to the exponent. Font size alone doesn't explain why $a + b^c$ means $a + (b^c)$ and not $(a + b)^c$ and the same for $a\cdot b^c$ vs. $a\cdot(b^c)$ respectively $(a\cdot b)^c$. For these, I'd argue that it's also a matter of visual grouping. In both $a\cdot b^c$ and $a + b^c$, the exponent is written extremely close to the $b$, without a symbol which'd separate the two. On the other hand $a$ and $b$ are separated by either a $+$ or a $\cdot$. Now, for multiplication the dot may be omitted, but it doesn't have to be omitted, i.e. since $ab$ and $a\cdot b$ are equivalent, one naturally wants $ab^c$ and $a\cdot b^c$ to be equivalent too.
For multiplication, division, addition subtraction, I always felt that the choice is somewhat arbitrary. Having said that, one reason that does speak in favour of having multiplication take precedence over addition is that one is allowed to leave out the dot and simply write $ab$ instead of $a\cdot b$. Since this isn't allowed for addition, in a lot of cases the terms which are multiplied will be closer together than those which are added, so most people will probably recognize them as "belonging together".
You may then ask "how come we're allowed to leave out the dot, but not the plus sign". This, I believe is a leftover from times when equations where stated in natural language. In most langues, you say something like "three apples" to indicate, well, three apples. In other words, you simply prefix a thing by a number to indicate multiple instances of that thing. This property of natural languages is mimicked in equations by allowing one to write $3x$ with the understanding that it means "3 of whatever $x$ is".
Best Answer
Any parts of an expression grouped with grouping symbols should be evaluated first, followed by exponents and radicals, then multiplication and division, then addition and subtraction.
Grouping symbols may include parentheses/brackets, such as $()$ $[]$ $\{\}$, and vincula (singular vinculum), such as the horizontal bar in a fraction or the horizontal bar extending over the contents of a radical.
Multiple exponentiations in sequence are evaluated right-to-left ($a^{b^c}=a^{(b^c)}$, not $(a^b)^c=a^{bc}$).
It is commonly taught, though not necessarily standard, that ungrouped multiplication and division (or, similarly, addition and subtraction) should be evaluated from left to right. (The mnemonics PEMDAS and BEDMAS sometimes give students the idea that multiplication and division [or similarly, addition and subtraction] are evaluated in separate steps, rather than together at one step.)
Implied multiplication (multiplication indicated by juxtaposition rather than an actual multiplication symbol) and the use of a $/$ to indicate division often cause ambiguity (or at least difficulty in proper interpretation), as evidenced by the $48/2(9+3)$ or $48รท2(9+3)$ meme. This is exacerbated by the existence of calculators (notably the obsolete Texas Instruments TI-81 and TI-85), which (at least in some instances) treated the $/$ division symbol as if it were a vinculum, grouping everything after it.